Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.44.0-wmf.6 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk 4 28 2691654 2691596 2024-12-12T16:41:04Z Ottawahitech 2369270 /* 'Wikidata item' link is moving, finally. */ Reply warning: autocomplete may take over anytime 2691654 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&filter= {{int:please-translate}}]'' Dear all, The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now. '''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC. Please share this message with members of your community so they can participate as well. In cooperation with the U4C,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == User group for Wikiversians == Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC) :Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:20, 8 August 2024 (UTC) :It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: :https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager :What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC) ::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:05, 7 October 2024 (UTC) ::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC) :I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC) == Rich's ''Illustrated Companion'' at Wikiversity: Right place? == Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project. {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC) :Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:26, 17 August 2024 (UTC) ::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be "open", i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC) :::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere. :::Also, as nitpick, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:06, 17 August 2024 (UTC) ::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a "container" which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC) == Coming soon: A new sub-referencing feature – try it! == <section begin="Sub-referencing"/> [[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]] Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]]. '''We want your feedback''' to make sure this feature works well for you: * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. * [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities. [[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand. Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == New [[Template:Form]] == Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC) == Sign up for the language community meeting on August 30th, 15:00 UTC == Hi all, The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page]. This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites. Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation! [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Template consolidation: User talk page block notice == Wondering if someone who likes templates could have a go at consolidating or helping decide between use of: * [[Template:Block]] * [[Template:Blocked]] Unless I'm missing something, it seems like we don't need both? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:16, 23 August 2024 (UTC) : I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere). : My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC) == Announcing the Universal Code of Conduct Coordinating Committee == <section begin="announcement-content" /> :''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&filter= {{int:please-translate}}]'' Hello all, The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]]. I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026: * North America (USA and Canada) ** Ajraddatz The following seats were not filled during this special election: * Latin America and Caribbean * Central and East Europe (CEE) * Sub-Saharan Africa * South Asia * The four remaining Community-At-Large seats Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community. Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]]. On behalf of the U4C and the Elections Committee,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Re: The Vector 2022 skin as the default in two weeks? == [[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]] Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then. What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here. So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC) :Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:35, 13 September 2024 (UTC) * I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC) *:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone. *:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users. *:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify. *:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy. *:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC) *:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of "you can switch" surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC) == Have your say: Vote for the 2024 Board of Trustees! == <section begin="announcement-content" /> Hello all, The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board. Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]]. When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''. To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]]. Best regards, The Elections Committee and Board Selection Working Group<section end="announcement-content" /> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? Many thanks, [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC) :It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:17, 15 September 2024 (UTC) == I hereby request for your Unblocking IP address and just reviewed and received a reverted rec == Hi there. {{unsigned|Ishmael Raphasha}} :No one has any clue what you're talking about. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:53, 18 September 2024 (UTC) == RICH-2K: New project with some initial questions == Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow): * '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''? * '''(Technical) Structure''': I am planning to set up this project at ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories? * '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)? * '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories? I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC) :*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope. :*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that "RICH-2K" is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages. :*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources. :*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask. :I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 20 September 2024 (UTC) ::Hello Justin! ::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first). ::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought. ::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved. ::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so. ::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC) :::re: upload, I'm just suggesting your sandbox(es) as you asked about "a test-environment". Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:39, 23 September 2024 (UTC) Hello! I have two further questions: # I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project. # ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage. Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC) :To answer your questions here: :*No, you are not contravening any policies we have. :*A leading "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." is acceptable. :*An accent in a category title is acceptable. :I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually. :As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:18, 25 September 2024 (UTC) ::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles "on the map" (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC) == Your wiki will be in read-only soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. '''You will be able to read, but not edit, all wikis for a short period of time.''' *You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}. *If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': *Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{tracked|T66315}} Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project. The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> section. We would like the Wiki communities feedback so please let us know or ask questions on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2. More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].<br><br>We welcome your feedback and questions.<br> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:04, 3 October 2024 (UTC) == Protected template bug for Pp == It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]). This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC) :The problem is that "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more "Wikipedia"s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:55, 5 October 2024 (UTC) == Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey == Dear all, We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community. To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event. Survey link: https://forms.gle/f66MuzjcPpwzVymu5 Please take a few minutes to share your thoughts. Your input will make a difference! Thank you for being a part of our journey to make Wiki Loves Ramadan a success. Warm regards, User:ZI Jony 03:19, 6 October 2024 (UTC) Wiki Loves Ramadan Organizing Team <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == 'Edit to my talk page' notification bug? == This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC) :I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:39, 7 October 2024 (UTC) ::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC) :::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:57, 7 October 2024 (UTC) ::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC) == [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] == Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC) :Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 8 October 2024 (UTC) ::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC) :::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:21, 8 October 2024 (UTC) : Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: "Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information." --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC) ::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason. ::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. ::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC) ::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC) ::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC) ::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to "Information sciences" (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC) ::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC) == [[:Category:Occupational Epidemiology]] == I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems <code>|filing=deep</code> would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC) : I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC) ::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC) == Active editors == It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC) :Odd. Maybe related to the school year? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 9 October 2024 (UTC) ::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC) :::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 9 October 2024 (UTC) ::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC) :::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC) == Intentionally incorrect resource == There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that: <blockquote>I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.</blockquote> However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC) :Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:38, 9 October 2024 (UTC) == [[:Category:Proposed guidelines]] == Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC) == [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] == Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC) : I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC) ::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:39, 13 October 2024 (UTC) ::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned "Wikiversal" package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC) == Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections == <section begin="announcement-content" /> Hello all, Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted. The following four candidates were the most voted: # [[User:Kritzolina|Christel Steigenberger]] # [[User:Nadzik|Maciej Artur Nadzikiewicz]] # [[User:Victoria|Victoria Doronina]] # [[User:Laurentius|Lorenzo Losa]] While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024. [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]] Best regards, The Elections Committee and Board Selection Working Group <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> Each year, typically from October through December, several of the movement’s committees seek new volunteers. Read more about the committees on their Meta-wiki pages: * [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]] * [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]] * [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]] Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have. For the Committee Support team, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == Interactive elements == Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea. [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC) : This is beyond my technical knowledge, but have you checked out: :* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript? :* [[Wikipedia:WikiProject JavaScript]] :* [[MediaWiki:Common.js]] :What sort of interactive elements are you thinking about? : Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 18 October 2024 (UTC) ::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC) :::From a quick look, maybe check out: :::* [[mw:Extension:Graph]] :::* [[phab:tag/graphs]] :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:40, 18 October 2024 (UTC) :::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC) == An unexplained spurt of Wikiversity page views == The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC) :Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC) == Center tempate failed on a contributors phone... == See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&diff=prev&oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC) == Essay-like page in user space that makes little sense and seems incoherent == The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC) == 'Wikidata item' link is moving, finally. == Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> :Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely. :I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC) ::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC) :::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list. :::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC) == Final Reminder: Join us in Making Wiki Loves Ramadan Success == Dear all, We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community. Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs. * Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey] * Deadline: November 10, 2024 Please take a few minutes to share your thoughts. Your input will truly make a difference! '''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply. * Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here] * Application Deadline: October 31, 2024 Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions] Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success! Warm regards,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == Android app for Wikiversity == Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC) :Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) :@[[User:Gryllida|Gryllida]]: Would you explain your terminology for those of us not in the know. What does ''push notifications'' mean? I use [[notifications]] when I am communicating onwikimediaprojects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC) :Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:23, 3 November 2024 (UTC) ::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:31, 3 November 2024 (UTC) == Language translation requests? == Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC) == Sign up for the language community meeting on November 29th, 16:00 UTC == Hello everyone, The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation. Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC) :Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC) ::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC) :::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC) :Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not "professors and scientists". Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC) ::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC) :::because I'm personally not a "professor" or a "scientist" and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) :@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious. :I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC) ::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC) jgwgtb911sswandwebziux646gn79jn 2691658 2691654 2024-12-12T16:51:16Z Ottawahitech 2369270 /* Wikiversity - Newsletters */ Reply 2691658 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&filter= {{int:please-translate}}]'' Dear all, The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now. '''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC. Please share this message with members of your community so they can participate as well. In cooperation with the U4C,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == User group for Wikiversians == Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC) :Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:20, 8 August 2024 (UTC) :It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: :https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager :What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC) ::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:05, 7 October 2024 (UTC) ::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC) :I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC) == Rich's ''Illustrated Companion'' at Wikiversity: Right place? == Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project. {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC) :Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:26, 17 August 2024 (UTC) ::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be "open", i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC) :::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere. :::Also, as nitpick, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:06, 17 August 2024 (UTC) ::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a "container" which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC) == Coming soon: A new sub-referencing feature – try it! == <section begin="Sub-referencing"/> [[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]] Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]]. '''We want your feedback''' to make sure this feature works well for you: * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. * [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities. [[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand. Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == New [[Template:Form]] == Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC) == Sign up for the language community meeting on August 30th, 15:00 UTC == Hi all, The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page]. This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites. Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation! [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Template consolidation: User talk page block notice == Wondering if someone who likes templates could have a go at consolidating or helping decide between use of: * [[Template:Block]] * [[Template:Blocked]] Unless I'm missing something, it seems like we don't need both? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:16, 23 August 2024 (UTC) : I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere). : My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC) == Announcing the Universal Code of Conduct Coordinating Committee == <section begin="announcement-content" /> :''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&filter= {{int:please-translate}}]'' Hello all, The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]]. I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026: * North America (USA and Canada) ** Ajraddatz The following seats were not filled during this special election: * Latin America and Caribbean * Central and East Europe (CEE) * Sub-Saharan Africa * South Asia * The four remaining Community-At-Large seats Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community. Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]]. On behalf of the U4C and the Elections Committee,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Re: The Vector 2022 skin as the default in two weeks? == [[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]] Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then. What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here. So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC) :Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:35, 13 September 2024 (UTC) * I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC) *:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone. *:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users. *:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify. *:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy. *:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC) *:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of "you can switch" surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC) == Have your say: Vote for the 2024 Board of Trustees! == <section begin="announcement-content" /> Hello all, The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board. Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]]. When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''. To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]]. Best regards, The Elections Committee and Board Selection Working Group<section end="announcement-content" /> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? Many thanks, [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC) :It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:17, 15 September 2024 (UTC) == I hereby request for your Unblocking IP address and just reviewed and received a reverted rec == Hi there. {{unsigned|Ishmael Raphasha}} :No one has any clue what you're talking about. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:53, 18 September 2024 (UTC) == RICH-2K: New project with some initial questions == Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow): * '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''? * '''(Technical) Structure''': I am planning to set up this project at ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories? * '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)? * '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories? I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC) :*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope. :*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that "RICH-2K" is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages. :*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources. :*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask. :I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 20 September 2024 (UTC) ::Hello Justin! ::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first). ::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought. ::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved. ::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so. ::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC) :::re: upload, I'm just suggesting your sandbox(es) as you asked about "a test-environment". Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:39, 23 September 2024 (UTC) Hello! I have two further questions: # I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project. # ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage. Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC) :To answer your questions here: :*No, you are not contravening any policies we have. :*A leading "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." is acceptable. :*An accent in a category title is acceptable. :I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually. :As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:18, 25 September 2024 (UTC) ::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles "on the map" (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC) == Your wiki will be in read-only soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. '''You will be able to read, but not edit, all wikis for a short period of time.''' *You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}. *If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': *Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{tracked|T66315}} Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project. The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> section. We would like the Wiki communities feedback so please let us know or ask questions on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2. More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].<br><br>We welcome your feedback and questions.<br> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:04, 3 October 2024 (UTC) == Protected template bug for Pp == It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]). This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC) :The problem is that "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more "Wikipedia"s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:55, 5 October 2024 (UTC) == Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey == Dear all, We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community. To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event. Survey link: https://forms.gle/f66MuzjcPpwzVymu5 Please take a few minutes to share your thoughts. Your input will make a difference! Thank you for being a part of our journey to make Wiki Loves Ramadan a success. Warm regards, User:ZI Jony 03:19, 6 October 2024 (UTC) Wiki Loves Ramadan Organizing Team <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == 'Edit to my talk page' notification bug? == This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC) :I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:39, 7 October 2024 (UTC) ::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC) :::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:57, 7 October 2024 (UTC) ::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC) == [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] == Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC) :Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 8 October 2024 (UTC) ::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC) :::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:21, 8 October 2024 (UTC) : Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: "Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information." --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC) ::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason. ::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. ::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC) ::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC) ::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC) ::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to "Information sciences" (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC) ::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC) == [[:Category:Occupational Epidemiology]] == I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems <code>|filing=deep</code> would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC) : I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC) ::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC) == Active editors == It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC) :Odd. Maybe related to the school year? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 9 October 2024 (UTC) ::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC) :::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 9 October 2024 (UTC) ::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC) :::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC) == Intentionally incorrect resource == There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that: <blockquote>I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.</blockquote> However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC) :Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:38, 9 October 2024 (UTC) == [[:Category:Proposed guidelines]] == Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC) == [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] == Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC) : I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC) ::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:39, 13 October 2024 (UTC) ::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned "Wikiversal" package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC) == Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections == <section begin="announcement-content" /> Hello all, Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted. The following four candidates were the most voted: # [[User:Kritzolina|Christel Steigenberger]] # [[User:Nadzik|Maciej Artur Nadzikiewicz]] # [[User:Victoria|Victoria Doronina]] # [[User:Laurentius|Lorenzo Losa]] While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024. [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]] Best regards, The Elections Committee and Board Selection Working Group <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> Each year, typically from October through December, several of the movement’s committees seek new volunteers. Read more about the committees on their Meta-wiki pages: * [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]] * [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]] * [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]] Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have. For the Committee Support team, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == Interactive elements == Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea. [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC) : This is beyond my technical knowledge, but have you checked out: :* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript? :* [[Wikipedia:WikiProject JavaScript]] :* [[MediaWiki:Common.js]] :What sort of interactive elements are you thinking about? : Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 18 October 2024 (UTC) ::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC) :::From a quick look, maybe check out: :::* [[mw:Extension:Graph]] :::* [[phab:tag/graphs]] :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:40, 18 October 2024 (UTC) :::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC) == An unexplained spurt of Wikiversity page views == The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC) :Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC) == Center tempate failed on a contributors phone... == See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&diff=prev&oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC) == Essay-like page in user space that makes little sense and seems incoherent == The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC) == 'Wikidata item' link is moving, finally. == Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> :Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely. :I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC) ::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC) :::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list. :::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC) == Final Reminder: Join us in Making Wiki Loves Ramadan Success == Dear all, We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community. Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs. * Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey] * Deadline: November 10, 2024 Please take a few minutes to share your thoughts. Your input will truly make a difference! '''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply. * Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here] * Application Deadline: October 31, 2024 Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions] Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success! Warm regards,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == Android app for Wikiversity == Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC) :Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) :@[[User:Gryllida|Gryllida]]: Would you explain your terminology for those of us not in the know. What does ''push notifications'' mean? I use [[notifications]] when I am communicating onwikimediaprojects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC) :Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:23, 3 November 2024 (UTC) ::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:31, 3 November 2024 (UTC) == Language translation requests? == Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC) == Sign up for the language community meeting on November 29th, 16:00 UTC == Hello everyone, The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation. Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC) :Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC) ::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC) :::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC) :Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not "professors and scientists". Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC) ::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC) :::because I'm personally not a "professor" or a "scientist" and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) :@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious. :I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC) ::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC) :::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? :::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter? :::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC) ldjgghr9znh92f2v7l7upxd1bvis8x2 2691660 2691658 2024-12-12T16:53:22Z RockTransport 2992610 /* Wikiversity - Newsletters */ Reply 2691660 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&filter= {{int:please-translate}}]'' Dear all, The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now. '''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC. Please share this message with members of your community so they can participate as well. In cooperation with the U4C,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == User group for Wikiversians == Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC) :Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:20, 8 August 2024 (UTC) :It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: :https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager :What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC) ::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:05, 7 October 2024 (UTC) ::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC) :I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC) == Rich's ''Illustrated Companion'' at Wikiversity: Right place? == Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project. {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC) :Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:26, 17 August 2024 (UTC) ::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be "open", i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC) :::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere. :::Also, as nitpick, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:06, 17 August 2024 (UTC) ::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a "container" which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC) == Coming soon: A new sub-referencing feature – try it! == <section begin="Sub-referencing"/> [[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]] Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]]. '''We want your feedback''' to make sure this feature works well for you: * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. * [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities. [[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand. Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == New [[Template:Form]] == Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC) == Sign up for the language community meeting on August 30th, 15:00 UTC == Hi all, The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page]. This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites. Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation! [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Template consolidation: User talk page block notice == Wondering if someone who likes templates could have a go at consolidating or helping decide between use of: * [[Template:Block]] * [[Template:Blocked]] Unless I'm missing something, it seems like we don't need both? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:16, 23 August 2024 (UTC) : I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere). : My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC) == Announcing the Universal Code of Conduct Coordinating Committee == <section begin="announcement-content" /> :''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&filter= {{int:please-translate}}]'' Hello all, The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]]. I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026: * North America (USA and Canada) ** Ajraddatz The following seats were not filled during this special election: * Latin America and Caribbean * Central and East Europe (CEE) * Sub-Saharan Africa * South Asia * The four remaining Community-At-Large seats Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community. Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]]. On behalf of the U4C and the Elections Committee,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Re: The Vector 2022 skin as the default in two weeks? == [[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]] Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then. What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here. So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC) :Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:35, 13 September 2024 (UTC) * I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC) *:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone. *:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users. *:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify. *:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy. *:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC) *:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of "you can switch" surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC) == Have your say: Vote for the 2024 Board of Trustees! == <section begin="announcement-content" /> Hello all, The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board. Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]]. When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''. To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]]. Best regards, The Elections Committee and Board Selection Working Group<section end="announcement-content" /> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? Many thanks, [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC) :It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:17, 15 September 2024 (UTC) == I hereby request for your Unblocking IP address and just reviewed and received a reverted rec == Hi there. {{unsigned|Ishmael Raphasha}} :No one has any clue what you're talking about. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:53, 18 September 2024 (UTC) == RICH-2K: New project with some initial questions == Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow): * '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''? * '''(Technical) Structure''': I am planning to set up this project at ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories? * '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)? * '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories? I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC) :*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope. :*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that "RICH-2K" is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages. :*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources. :*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask. :I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 20 September 2024 (UTC) ::Hello Justin! ::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first). ::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought. ::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved. ::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so. ::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC) :::re: upload, I'm just suggesting your sandbox(es) as you asked about "a test-environment". Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:39, 23 September 2024 (UTC) Hello! I have two further questions: # I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project. # ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage. Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC) :To answer your questions here: :*No, you are not contravening any policies we have. :*A leading "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." is acceptable. :*An accent in a category title is acceptable. :I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually. :As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:18, 25 September 2024 (UTC) ::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles "on the map" (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC) == Your wiki will be in read-only soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. '''You will be able to read, but not edit, all wikis for a short period of time.''' *You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}. *If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': *Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{tracked|T66315}} Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project. The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> section. We would like the Wiki communities feedback so please let us know or ask questions on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2. More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].<br><br>We welcome your feedback and questions.<br> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:04, 3 October 2024 (UTC) == Protected template bug for Pp == It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]). This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC) :The problem is that "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more "Wikipedia"s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:55, 5 October 2024 (UTC) == Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey == Dear all, We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community. To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event. Survey link: https://forms.gle/f66MuzjcPpwzVymu5 Please take a few minutes to share your thoughts. Your input will make a difference! Thank you for being a part of our journey to make Wiki Loves Ramadan a success. Warm regards, User:ZI Jony 03:19, 6 October 2024 (UTC) Wiki Loves Ramadan Organizing Team <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == 'Edit to my talk page' notification bug? == This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC) :I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:39, 7 October 2024 (UTC) ::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC) :::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:57, 7 October 2024 (UTC) ::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC) == [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] == Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC) :Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 8 October 2024 (UTC) ::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC) :::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:21, 8 October 2024 (UTC) : Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: "Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information." --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC) ::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason. ::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. ::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC) ::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC) ::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC) ::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to "Information sciences" (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC) ::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC) == [[:Category:Occupational Epidemiology]] == I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems <code>|filing=deep</code> would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC) : I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC) ::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC) == Active editors == It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC) :Odd. Maybe related to the school year? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 9 October 2024 (UTC) ::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC) :::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 9 October 2024 (UTC) ::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC) :::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC) == Intentionally incorrect resource == There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that: <blockquote>I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.</blockquote> However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC) :Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:38, 9 October 2024 (UTC) == [[:Category:Proposed guidelines]] == Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC) == [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] == Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC) : I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC) ::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:39, 13 October 2024 (UTC) ::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned "Wikiversal" package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC) == Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections == <section begin="announcement-content" /> Hello all, Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted. The following four candidates were the most voted: # [[User:Kritzolina|Christel Steigenberger]] # [[User:Nadzik|Maciej Artur Nadzikiewicz]] # [[User:Victoria|Victoria Doronina]] # [[User:Laurentius|Lorenzo Losa]] While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024. [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]] Best regards, The Elections Committee and Board Selection Working Group <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> Each year, typically from October through December, several of the movement’s committees seek new volunteers. Read more about the committees on their Meta-wiki pages: * [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]] * [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]] * [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]] Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have. For the Committee Support team, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == Interactive elements == Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea. [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC) : This is beyond my technical knowledge, but have you checked out: :* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript? :* [[Wikipedia:WikiProject JavaScript]] :* [[MediaWiki:Common.js]] :What sort of interactive elements are you thinking about? : Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 18 October 2024 (UTC) ::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC) :::From a quick look, maybe check out: :::* [[mw:Extension:Graph]] :::* [[phab:tag/graphs]] :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:40, 18 October 2024 (UTC) :::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC) == An unexplained spurt of Wikiversity page views == The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC) :Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC) == Center tempate failed on a contributors phone... == See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&diff=prev&oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC) == Essay-like page in user space that makes little sense and seems incoherent == The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC) == 'Wikidata item' link is moving, finally. == Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> :Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely. :I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC) ::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC) :::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list. :::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC) == Final Reminder: Join us in Making Wiki Loves Ramadan Success == Dear all, We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community. Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs. * Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey] * Deadline: November 10, 2024 Please take a few minutes to share your thoughts. Your input will truly make a difference! '''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply. * Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here] * Application Deadline: October 31, 2024 Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions] Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success! Warm regards,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == Android app for Wikiversity == Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC) :Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) :@[[User:Gryllida|Gryllida]]: Would you explain your terminology for those of us not in the know. What does ''push notifications'' mean? I use [[notifications]] when I am communicating onwikimediaprojects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC) :Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:23, 3 November 2024 (UTC) ::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:31, 3 November 2024 (UTC) == Language translation requests? == Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC) == Sign up for the language community meeting on November 29th, 16:00 UTC == Hello everyone, The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation. Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC) :Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC) ::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC) :::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC) :Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not "professors and scientists". Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC) ::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC) :::because I'm personally not a "professor" or a "scientist" and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) :@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious. :I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC) ::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC) :::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? :::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter? :::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC) ::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC) 7ag4b187rsv8jzejtc55uve90vldta7 2691662 2691660 2024-12-12T17:00:37Z Ottawahitech 2369270 /* Android app for Wikiversity */ blurry notifications 2691662 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&filter= {{int:please-translate}}]'' Dear all, The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now. '''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC. Please share this message with members of your community so they can participate as well. In cooperation with the U4C,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == User group for Wikiversians == Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC) :Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:20, 8 August 2024 (UTC) :It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: :https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager :What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC) ::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:05, 7 October 2024 (UTC) ::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC) :I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC) == Rich's ''Illustrated Companion'' at Wikiversity: Right place? == Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project. {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC) :Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:26, 17 August 2024 (UTC) ::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be "open", i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC) :::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere. :::Also, as nitpick, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:06, 17 August 2024 (UTC) ::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a "container" which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC) == Coming soon: A new sub-referencing feature – try it! == <section begin="Sub-referencing"/> [[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]] Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]]. '''We want your feedback''' to make sure this feature works well for you: * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. * [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities. [[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand. Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == New [[Template:Form]] == Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC) == Sign up for the language community meeting on August 30th, 15:00 UTC == Hi all, The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page]. This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites. Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation! [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Template consolidation: User talk page block notice == Wondering if someone who likes templates could have a go at consolidating or helping decide between use of: * [[Template:Block]] * [[Template:Blocked]] Unless I'm missing something, it seems like we don't need both? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:16, 23 August 2024 (UTC) : I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere). : My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC) == Announcing the Universal Code of Conduct Coordinating Committee == <section begin="announcement-content" /> :''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&filter= {{int:please-translate}}]'' Hello all, The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]]. I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026: * North America (USA and Canada) ** Ajraddatz The following seats were not filled during this special election: * Latin America and Caribbean * Central and East Europe (CEE) * Sub-Saharan Africa * South Asia * The four remaining Community-At-Large seats Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community. Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]]. On behalf of the U4C and the Elections Committee,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Re: The Vector 2022 skin as the default in two weeks? == [[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]] Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then. What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here. So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC) :Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:35, 13 September 2024 (UTC) * I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC) *:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone. *:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users. *:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify. *:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy. *:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC) *:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of "you can switch" surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC) == Have your say: Vote for the 2024 Board of Trustees! == <section begin="announcement-content" /> Hello all, The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board. Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]]. When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''. To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]]. Best regards, The Elections Committee and Board Selection Working Group<section end="announcement-content" /> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? Many thanks, [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC) :It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:17, 15 September 2024 (UTC) == I hereby request for your Unblocking IP address and just reviewed and received a reverted rec == Hi there. {{unsigned|Ishmael Raphasha}} :No one has any clue what you're talking about. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:53, 18 September 2024 (UTC) == RICH-2K: New project with some initial questions == Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow): * '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''? * '''(Technical) Structure''': I am planning to set up this project at ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories? * '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)? * '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories? I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC) :*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope. :*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that "RICH-2K" is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages. :*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources. :*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask. :I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 20 September 2024 (UTC) ::Hello Justin! ::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first). ::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought. ::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved. ::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so. ::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC) :::re: upload, I'm just suggesting your sandbox(es) as you asked about "a test-environment". Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:39, 23 September 2024 (UTC) Hello! I have two further questions: # I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project. # ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage. Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC) :To answer your questions here: :*No, you are not contravening any policies we have. :*A leading "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." is acceptable. :*An accent in a category title is acceptable. :I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually. :As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:18, 25 September 2024 (UTC) ::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles "on the map" (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC) == Your wiki will be in read-only soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. '''You will be able to read, but not edit, all wikis for a short period of time.''' *You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}. *If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': *Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{tracked|T66315}} Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project. The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> section. We would like the Wiki communities feedback so please let us know or ask questions on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2. More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].<br><br>We welcome your feedback and questions.<br> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:04, 3 October 2024 (UTC) == Protected template bug for Pp == It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]). This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC) :The problem is that "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more "Wikipedia"s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:55, 5 October 2024 (UTC) == Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey == Dear all, We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community. To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event. Survey link: https://forms.gle/f66MuzjcPpwzVymu5 Please take a few minutes to share your thoughts. Your input will make a difference! Thank you for being a part of our journey to make Wiki Loves Ramadan a success. Warm regards, User:ZI Jony 03:19, 6 October 2024 (UTC) Wiki Loves Ramadan Organizing Team <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == 'Edit to my talk page' notification bug? == This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC) :I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:39, 7 October 2024 (UTC) ::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC) :::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:57, 7 October 2024 (UTC) ::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC) == [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] == Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC) :Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 8 October 2024 (UTC) ::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC) :::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:21, 8 October 2024 (UTC) : Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: "Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information." --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC) ::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason. ::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. ::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC) ::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC) ::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC) ::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to "Information sciences" (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC) ::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC) == [[:Category:Occupational Epidemiology]] == I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems <code>|filing=deep</code> would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC) : I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC) ::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC) == Active editors == It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC) :Odd. Maybe related to the school year? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 9 October 2024 (UTC) ::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC) :::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 9 October 2024 (UTC) ::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC) :::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC) == Intentionally incorrect resource == There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that: <blockquote>I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.</blockquote> However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC) :Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:38, 9 October 2024 (UTC) == [[:Category:Proposed guidelines]] == Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC) == [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] == Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC) : I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC) ::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:39, 13 October 2024 (UTC) ::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned "Wikiversal" package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC) == Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections == <section begin="announcement-content" /> Hello all, Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted. The following four candidates were the most voted: # [[User:Kritzolina|Christel Steigenberger]] # [[User:Nadzik|Maciej Artur Nadzikiewicz]] # [[User:Victoria|Victoria Doronina]] # [[User:Laurentius|Lorenzo Losa]] While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024. [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]] Best regards, The Elections Committee and Board Selection Working Group <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> Each year, typically from October through December, several of the movement’s committees seek new volunteers. Read more about the committees on their Meta-wiki pages: * [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]] * [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]] * [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]] Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have. For the Committee Support team, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == Interactive elements == Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea. [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC) : This is beyond my technical knowledge, but have you checked out: :* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript? :* [[Wikipedia:WikiProject JavaScript]] :* [[MediaWiki:Common.js]] :What sort of interactive elements are you thinking about? : Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 18 October 2024 (UTC) ::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC) :::From a quick look, maybe check out: :::* [[mw:Extension:Graph]] :::* [[phab:tag/graphs]] :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:40, 18 October 2024 (UTC) :::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC) == An unexplained spurt of Wikiversity page views == The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC) :Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC) == Center tempate failed on a contributors phone... == See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&diff=prev&oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC) == Essay-like page in user space that makes little sense and seems incoherent == The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC) == 'Wikidata item' link is moving, finally. == Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> :Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely. :I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC) ::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC) :::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list. :::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC) == Final Reminder: Join us in Making Wiki Loves Ramadan Success == Dear all, We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community. Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs. * Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey] * Deadline: November 10, 2024 Please take a few minutes to share your thoughts. Your input will truly make a difference! '''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply. * Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here] * Application Deadline: October 31, 2024 Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions] Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success! Warm regards,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == Android app for Wikiversity == Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC) :Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) :@[[User:Gryllida|Gryllida]]: Would you explain your terminology for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC) :Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:23, 3 November 2024 (UTC) ::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:31, 3 November 2024 (UTC) == Language translation requests? == Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC) == Sign up for the language community meeting on November 29th, 16:00 UTC == Hello everyone, The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation. Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC) :Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC) ::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC) :::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC) :Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not "professors and scientists". Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC) ::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC) :::because I'm personally not a "professor" or a "scientist" and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) :@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious. :I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC) ::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC) :::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? :::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter? :::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC) ::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC) ox0z85wcgvdao3hj5ssmfhp6wj49zh7 2691664 2691662 2024-12-12T17:05:31Z Ottawahitech 2369270 /* Events on Wikiversity */ Reply warning: autocomplete may take over anytime 2691664 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&filter= {{int:please-translate}}]'' Dear all, The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now. '''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC. Please share this message with members of your community so they can participate as well. In cooperation with the U4C,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == User group for Wikiversians == Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC) :Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:20, 8 August 2024 (UTC) :It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: :https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager :What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC) ::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:05, 7 October 2024 (UTC) ::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC) :I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC) == Rich's ''Illustrated Companion'' at Wikiversity: Right place? == Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project. {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC) :Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:26, 17 August 2024 (UTC) ::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be "open", i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC) :::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere. :::Also, as nitpick, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:06, 17 August 2024 (UTC) ::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a "container" which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC) == Coming soon: A new sub-referencing feature – try it! == <section begin="Sub-referencing"/> [[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]] Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]]. '''We want your feedback''' to make sure this feature works well for you: * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. * [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities. [[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand. Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == New [[Template:Form]] == Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC) == Sign up for the language community meeting on August 30th, 15:00 UTC == Hi all, The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page]. This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites. Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation! [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Template consolidation: User talk page block notice == Wondering if someone who likes templates could have a go at consolidating or helping decide between use of: * [[Template:Block]] * [[Template:Blocked]] Unless I'm missing something, it seems like we don't need both? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:16, 23 August 2024 (UTC) : I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere). : My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC) == Announcing the Universal Code of Conduct Coordinating Committee == <section begin="announcement-content" /> :''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&filter= {{int:please-translate}}]'' Hello all, The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]]. I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026: * North America (USA and Canada) ** Ajraddatz The following seats were not filled during this special election: * Latin America and Caribbean * Central and East Europe (CEE) * Sub-Saharan Africa * South Asia * The four remaining Community-At-Large seats Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community. Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]]. On behalf of the U4C and the Elections Committee,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Re: The Vector 2022 skin as the default in two weeks? == [[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]] Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then. What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here. So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC) :Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:35, 13 September 2024 (UTC) * I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC) *:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone. *:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users. *:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify. *:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy. *:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC) *:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of "you can switch" surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC) == Have your say: Vote for the 2024 Board of Trustees! == <section begin="announcement-content" /> Hello all, The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board. Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]]. When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''. To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]]. Best regards, The Elections Committee and Board Selection Working Group<section end="announcement-content" /> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? Many thanks, [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC) :It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:17, 15 September 2024 (UTC) == I hereby request for your Unblocking IP address and just reviewed and received a reverted rec == Hi there. {{unsigned|Ishmael Raphasha}} :No one has any clue what you're talking about. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:53, 18 September 2024 (UTC) == RICH-2K: New project with some initial questions == Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow): * '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''? * '''(Technical) Structure''': I am planning to set up this project at ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories? * '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)? * '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories? I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC) :*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope. :*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that "RICH-2K" is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages. :*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources. :*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask. :I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 20 September 2024 (UTC) ::Hello Justin! ::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first). ::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought. ::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved. ::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so. ::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC) :::re: upload, I'm just suggesting your sandbox(es) as you asked about "a test-environment". Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:39, 23 September 2024 (UTC) Hello! I have two further questions: # I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project. # ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage. Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC) :To answer your questions here: :*No, you are not contravening any policies we have. :*A leading "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." is acceptable. :*An accent in a category title is acceptable. :I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually. :As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:18, 25 September 2024 (UTC) ::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles "on the map" (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC) == Your wiki will be in read-only soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. '''You will be able to read, but not edit, all wikis for a short period of time.''' *You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}. *If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': *Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{tracked|T66315}} Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project. The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> section. We would like the Wiki communities feedback so please let us know or ask questions on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2. More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].<br><br>We welcome your feedback and questions.<br> [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:04, 3 October 2024 (UTC) == Protected template bug for Pp == It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]). This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC) :The problem is that "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more "Wikipedia"s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:55, 5 October 2024 (UTC) == Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey == Dear all, We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community. To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event. Survey link: https://forms.gle/f66MuzjcPpwzVymu5 Please take a few minutes to share your thoughts. Your input will make a difference! Thank you for being a part of our journey to make Wiki Loves Ramadan a success. Warm regards, User:ZI Jony 03:19, 6 October 2024 (UTC) Wiki Loves Ramadan Organizing Team <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == 'Edit to my talk page' notification bug? == This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC) :I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:39, 7 October 2024 (UTC) ::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC) :::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 09:57, 7 October 2024 (UTC) ::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC) == [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] == Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC) :Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 8 October 2024 (UTC) ::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC) :::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:21, 8 October 2024 (UTC) : Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: "Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information." --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC) ::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason. ::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. ::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC) ::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC) ::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC) ::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to "Information sciences" (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC) ::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC) == [[:Category:Occupational Epidemiology]] == I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems <code>|filing=deep</code> would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC) : I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC) ::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC) == Active editors == It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC) :Odd. Maybe related to the school year? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 9 October 2024 (UTC) ::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC) :::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 9 October 2024 (UTC) ::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC) :::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC) == Intentionally incorrect resource == There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that: <blockquote>I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.</blockquote> However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC) :Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:38, 9 October 2024 (UTC) == [[:Category:Proposed guidelines]] == Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC) == [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] == Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC) : I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC) ::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:39, 13 October 2024 (UTC) ::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned "Wikiversal" package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC) == Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections == <section begin="announcement-content" /> Hello all, Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted. The following four candidates were the most voted: # [[User:Kritzolina|Christel Steigenberger]] # [[User:Nadzik|Maciej Artur Nadzikiewicz]] # [[User:Victoria|Victoria Doronina]] # [[User:Laurentius|Lorenzo Losa]] While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024. [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]] Best regards, The Elections Committee and Board Selection Working Group <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> Each year, typically from October through December, several of the movement’s committees seek new volunteers. Read more about the committees on their Meta-wiki pages: * [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]] * [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]] * [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]] Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have. For the Committee Support team, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == Interactive elements == Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea. [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC) : This is beyond my technical knowledge, but have you checked out: :* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript? :* [[Wikipedia:WikiProject JavaScript]] :* [[MediaWiki:Common.js]] :What sort of interactive elements are you thinking about? : Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 18 October 2024 (UTC) ::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC) :::From a quick look, maybe check out: :::* [[mw:Extension:Graph]] :::* [[phab:tag/graphs]] :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:40, 18 October 2024 (UTC) :::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC) == An unexplained spurt of Wikiversity page views == The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC) :Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC) == Center tempate failed on a contributors phone... == See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&diff=prev&oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC) == Essay-like page in user space that makes little sense and seems incoherent == The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC) == 'Wikidata item' link is moving, finally. == Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> :Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely. :I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC) ::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC) :::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list. :::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC) == Final Reminder: Join us in Making Wiki Loves Ramadan Success == Dear all, We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community. Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs. * Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey] * Deadline: November 10, 2024 Please take a few minutes to share your thoughts. Your input will truly make a difference! '''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply. * Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here] * Application Deadline: October 31, 2024 Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions] Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success! Warm regards,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == Android app for Wikiversity == Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC) :Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) :@[[User:Gryllida|Gryllida]]: Would you explain your terminology for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC) :Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:23, 3 November 2024 (UTC) ::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:31, 3 November 2024 (UTC) == Language translation requests? == Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC) == Sign up for the language community meeting on November 29th, 16:00 UTC == Hello everyone, The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation. Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC) :Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC) ::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC) :::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC) :Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not "professors and scientists". Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC) ::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC) :::because I'm personally not a "professor" or a "scientist" and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC) ::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) :@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious. :I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC) ::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC) :::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? :::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter? :::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC) ::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC) 7k6vghv3kfcaylxhtxby5qj78wc0j18 102 3781 2691685 2618709 2024-12-12T20:14:19Z DerGeist4040 2994919 2691685 wikitext text/x-wiki * SecretlyHistoric, interested in ancient history<!-- Add your name to the list using * ~~~~ --> * Adopt this portal by adding your name [[{{titleparts|2}}/Participants|here]] * [[User:Fothan|Fothan]] ([[User talk:Fothan|discuss]] • [[Special:Contributions/Fothan|contribs]]) 13:49, 30 August 2018 (UTC) * [[User:Msorre21|Msorre2]] * Tiamiyu Moses Oluwasegun * [[User:DErnestWachter|DErnestWachter]] * [[User:Mikeduke324|Mikeduke324]], interested in history. * [[User:सीमा1|सीमा1]], interested in Geography & hindi. * [[User:Pelanie For Life|Pelanie For Life]] ([[User talk:Pelanie For Life|discuss]] • [[Special:Contributions/Pelanie For Life|contribs]]) * Oliver Cremin, BA. Classics & Theology. * [[DerGeist4040]], interested in Greek and Roman history. 61w4yyknc0o9yssjy98pp97mf9k1tvh 0 9252 2691682 2632137 2024-12-12T19:54:25Z Dan Lock Author VR 2994930 2691682 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {{complete|text=This page is [[:Category:Completed resources|ready]] for students}}<br> {{Film School:Message|Style=Plain|Message=Enter the magical world of filmmaking.}} {| cellpadding="10" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: LemonChiffon; border: 1px solid Gray; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" | {{Courses in Filmmaking - Table - Float Right}} {{center top}}<br> {| border=0 cellspacing=0 cellpadding=5 | bgcolor="LemonChiffon" | [[Image:Crystal Clear app kfm home.png]] | bgcolor="LemonChiffon" | {{center top}}[[Film School/About|Wikiversity Film School]]<br><big>The Courses in<br>Narrative Film Production</big>{{center bottom}} |} {{center bottom}} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">[[Image:Crystal Clear action build.png|24px]] Click on the course name to begin the course. →</p> |} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">Note: These courses are only for dramatic motion pictures (movies with a script and dialog).<br>These courses are '''not''' for documentaries, event video, corporate video, educational programs, or multimedia.</p> {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 55%; background-color: Honeydew; border: 1px solid Gray; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="1" | [[Image:Nuvola apps edu miscellaneous.svg|right]] ===Anyone can learn to be a filmmaker=== ;Filmmaking is easy :Filmmaking is not rocket science. Everything about filmmaking is extremely easy to learn. Anyone can do it if they wish. ;The challenge of learning filmmaking :The challenge is filmmaking requires learning a huge number of skills. Each skill is easy to learn but the number of things you must learn is huge. :If you want to be an independent filmmaker, you must learn the equivalent of 20 different careers. Even if you are a fast learner, it can take you years to learn everything. ;Telling a story :In a dramatic motion picture, the story is told by many people. The [[wikipedia:Cinematographer|cinematographer]] tells the story with the camera. The lighting person tells the story with lighting. The film composer tells the story with music. The [[wikipedia:Actor|actors]] tell the story with action and dialog. The [[wikipedia:Film_editing|editor]] tells the story with editing. The [[wikipedia:Sound_design|sound designer]] tells the story with sound. ;You have to learn all of this :And as an independent filmmaker, you must learn all of these skills. :If you fail to learn even one of these skills, people will notice and be turned off by your movie. You must learn everything!!!! | style="width: 50%; background-color: #eef; border: 1px solid #777777; <!--vertical-align: top; -->-moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"; rowspan="3"| {{center top}}[[Image:Crystal_Clear_app_camera.png]]{{center bottom}} ==Film School Preparatory== :Wikiversity Film School is a {{font|color=Purple|'''preparatory school'''}} for budding filmmakers who plan to go to film school or take classes in motion picture production. :Each year, USC Film School receives 800 applicants to fill just 50 undergraduate positions. :We teach you the things that film schools expect you to know '''before''' you get to film school so that you can have a base to work from when you first start film-school. ==Learning filmmaking software== ;Intel processor :First, you need to do is learn about the computer software for filmmaking. See below for a list of the software you will need. Most is free! ;Designed for the Macintosh : The Film Scoring course is designed for Apple's GarageBand with Jam Pack<nowiki>:</nowiki> Symphony Orchestra :The Film Editing course is designed for Apple's Final Cut Pro or Adobe Premiere. ;Linux :Where possible, free Linux software is listed. :However, there is still no standard movie file format for Linux and nothing which matches the quality and usefulness of Apple's GarageBand for film scoring. :Therefore, Linux is currently not a good choice for filmmakers. |- | style="width: 50%; background-color: #fee; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"| [[Image:Nuvola apps kword.png|right|96px]] ==What do we do?== ;Make a tiny movie :In the filmmaking class, we create a short motion picture. It is less than a minute long so it is very simple. :But before you complete the animatic for the movie, you will also need to learn film [[Film editing|'''editing''']] and film [[Mad Max's Course in Film Scoring|'''scoring''']]. |- | style="width: 50%; background-color: #ffa; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"| [[Image:3D Universe Percy w cloths 1.png|right|90px]] ==Where to start?== ;Filmmaking, editing, and scoring :Budding filmmakers should start with the basic [[Filmmaking Basics|'''filmmaking course''']]. For the [[Filmmaking Basics/Formatting the Script|'''first lesson''']], you will format a very short motion picture script. After that, take the [[Filmmaking Basics/Thumbnail Pop Quiz|pop quiz]] where you tell me how you would begin to film this scene. :If you want to be a film composer (but you are not a musician), screen writer, or a film editor, you can go directly to the courses in [[Film editing|film editing]] or [[Mad Max's Course in Film Scoring|film scoring]] or the [[Filmmaking for High School Drama Departments|script writing exercise]]. :If you are a musician who wants to be a film composer, [[Film scoring/Introduction|click here]]. |} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">Note: These courses are designed for the Macintosh computer.<!--<br>PC World Magazines says, "The fastest Windows Vista notebook we've tested this year (through 10/25/07) is a Mac.<br>At 6.6 pounds and just 1 inch thick, the MacBook Pro is the lightest 17-inch notebook available."<br>Your instructor says, "Starting in 2008, if you are going to buy a new Windows PC, you might as well get a Macintosh.--> </p> {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 50%; background-color: #eef; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="1" | [[Image:Books-aj.svg_aj_ashton_01g.png|right|96px]] ==Cost== ;Lessons :These filmmaking courses are '''free'''. The only requirement is you submit your homework assignments so others can benefit. (see right) ;Software :I always try to select programs which are {{font|color=Green|'''free'''}}. | style="width: 50%; background-color: Honeydew; border: 1px solid #777777; <!--vertical-align: top; -->-moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"; rowspan="2"| <!--{{center top}}[[Image:Nuvola_apps_package_wordprocessing.png|96px]]{{center bottom}}--> [[Image:Nuvola_apps_package_wordprocessing.png|right|96px]] ==Sharing your work== ;Your homework assignments :Your completed assignments must be submitted under the Free Documentation License or as Public Domain. :This allows what you have learned to be shared by others. |} ---- ---- <p style = "font-size: 12pt;font-family: georgia; text-align: center; color: MidnightBlue">Free Software used at Wikiversity Film School</p> ---- ---- {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 100%; background-color: LemonChiffon; border: 2px solid Blue; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:Crystal Clear app package.png|right|128px]] =={{font|color=Green|'''Free'''}} Filmmaking Software at Wikiversity Film School== :There is some of free software used at Wikiversity Film School. Sometimes, this software is demo versions or simplified versions of software but this is good enough to complete these lessons. Using {{font|color=Green|'''free'''}} software, you can learn a tremendous amount about filmmaking. |- | style="width: 100%; background-color: #ffe; border: 2px solid Blue; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:FrameForge SBTDS Screen Shot.png|right|260px]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">FrameForge 3D Studio Free Demo Version</p> ;Basic filmmaking (pre-production) :The heart of the Wikiversity Film School's basic filmmaking course (pre-production) is [http://www.frameforge3d.com/download.php '''FrameForge 3D Studio 2 Demo Version''']. This is the most educational program used at Wikiversity Film School. And the demo version is {{font|color=Green|'''free'''}} for both the Macintosh and Windows operating system. :This program simulates the motion picture camera, the movie set, and the actors. The program is useful for experimenting with the different lenses on your motion picture camera. And the final output of FrameForge 3D Studio 2 Demo Version is completely frame accurate (including depth of field effects) ready to give your cinematographer for creating all the shots of your movie. :The {{font|color=Green|'''free'''}} demo version is limited to 20 uses. Other limitations apply. However, this is more than enough to complete your assignments at Wikiversity Film School. Download this '''free''' program today and begin learning how to use it. ---- == Redefining VR Motorbike Racing Games == Have you ever dreamed of feeling the rush of super bike racing from the comfort of your home? '''VRider SBK''' is here to make it a reality. This incredible '''SBK racing game''' takes virtual reality to a whole new level, offering an immersive experience that puts you right on the track. With lifelike physics, breathtaking graphics, and a variety of customisable bikes, it’s the ultimate game for fans of '''[https://vridergame.com/ VR motorbike games]'''. Whether you’re speeding through tight curves, competing in global tournaments, or enjoying multiplayer mode with friends, VRider SBK ensures every race feels real. The attention to detail is astounding, from the roar of the engines to the feel of the wind rushing past you. It’s not just a game—it’s an adventure for anyone who loves '''super bike games'''. == What do you think makes a great VR motorbike game? Let’s discuss your experiences with [https://vridergame.com/ super bike racing] in virtual reality and what features you’d like to see in future games. Share your thoughts and join the conversation! == [[Image:GarageBand Musical Typing.jpg|260px|right]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">Apple's GarageBand</p> ;Film Scoring :All film scoring lessons at Wikiversity Film School can be done using Apple's GarageBand which is {{font|color=Green|'''free'''}} with each new Macintosh PC. :In addition to a midi program like GarageBand, you will also need a good selection of musical instruments for the symphony orchestra. Some musical software instruments of the symphony orchestra for GarageBand are {{font|color=Green|'''free'''}} (such as the packages from Boldt) and some are not. I recommend Apple's Jam Pack:Symphony Orchestra. I have not tried Apple's new Jam Pack for voices which should also be useful. :*Filmmakers should {{font|color=OrangeRed|'''NOT'''}} use Apple's Logic 7 which is poorly designed and exceedingly awkward. I have not tried version 8 yet. :* I do not know what to recommend for Windows. ---- [[Image:The Assignment Screen Shot.png|right|140px]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">Film Dailies</p> ---- ;Animation :Starting in 2009, Wikiversity Film School will have a simple course in 3D animation. This will use the {{font|color=Green|'''free'''}} programs of DAZ Studio and Bryce 5.5. : If you have Poser and Vue, you can use those programs but they are not free. :DAZ Studio and Bryce take time to learn so if you are interested in working with 3D, you should download these {{font|color=Green|'''free'''}} programs and begin to learn them. ---- [[Image:ArtRage2Chalk100.jpg|right|200px]] ;Other useful programs :*[http://www.ambientdesign.com/artragedown.html ArtRage 2.5 Starter Edition] is a {{font|color=Green|free}} paint program used for matte painting and creating the movie poster. This is a simple and fun artistic painting program. Feels very natural. Extremely useful. If you need any kind of digital painting with traditional artist materials, this is your first choice. This is the {{font|color=Green|'''free'''}} version which is fully working but more limited than the full version which is also remarkably inexpensive. :* [https://gimp.en.softonic.com/mac GIMP] is a {{font|color=Green|'''free'''}} paint program comparable to Adobe Photoshop. Surprisingly mature program but definitely not as easy to use as Photoshop. Requires X-11. If you do not have Adobe Photoshop 4 or later, this program is absolutely necessary for some of the classes at Wikiversity Film School. :*[http://www.audacityteam.org/download/ Audacity] is a {{font|color=Green|'''free'''}} useful utility for working with audio and converting audio to OGG files. It is a bit awkward but it is free. :* [http://www.tuxpaint.org/ Tux Paint] is a {{font|color=Green|'''free'''}} fun paint program for kids. It has the added advantage of using rubber stamps and creating matte paintings easily. Has the unique feature of doing rough storyboards with special storyboard artwork. Also useful for one of the lessons about learning matte painting at Wikiversity Film School for kids. |} {| cellpadding="25" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 50%; background-color: #fed; border: 1px solid #777777; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="2" | [[Image:Crystal Clear app korganizer.png|128px|right]] ==Where to begin?== :*If you want to become a filmmaker, start with the '''[[Filmmaking Basics|Basic Filmmaking Course]]'''. [[Image:Mplayer.svg|32px]] :*Later, when you are ready, you should also take the '''[[Film editing|Film Editing Course]]'''. [[Image:Crystal Clear mimetype video.png|32px]] :*The most important course of all is film scoring. You will need the '''[[Film scoring| Film Scoring Course]]''' to complete the other two courses. [[Image:Crystal Clear app knotify.png|32px]] {{Contact your instructor: Filmmaking|page={{PAGENAME}}}} |} <!-- {{center top}}<small>[[Film school:site map|site map]] · [[Film school:Completed Homework Assignments|homework]]</small>{{center bottom}} --> ==External links== ;{{font|color=Red|NOT}} RECOMMENDED for these lessons :As far as I know, currently none of the programs listed below is compatible with the disks used in these lessons. -- ''Robert Elliott, your Instructor'' :* [http://www.musix.org.ar/en/index.html Musix] - A multimedia Linux distro (music and video) :* [http://jahshaka.org/ Jashaka] - comprehensive open source player and editor - "Powering the new Hollywood" :* [http://www.virtualdub.org/ VirtualDub] - Open source nonlinear editor with recent release :* [http://directory.fsf.org/category/vmanip/ FSF] - list of free video editing tools :* [http://fixounet.free.fr/avidemux/ Avidemux] - open source video editor * [http://digitaltippingpoint.com/wiki/index.php?title=Main_Page Digital Tipping Point wiki] - Collaborative open documentary ---- ---- * <small>Student in the Basic Filmmaking Course may [[Film school - Submitting student assignments - Basic filmmaking course|upload their own assignments]].</small> * <small>Student in the Film Scoring Course may [[Film school - Submitting student assignments - Film scoring course|upload their own assignments]].</small> * <small>Student in the Film Editing Course may [[Film school - Submitting student assignments - Film editing course|upload their own assignments]].</small> [[Category:{{PAGENAME}}| ]] [[Category:Narrative Film Editing]] [[Category:Filmmaking in the Drama Department]] [[Category:Film scoring]] [[Category:Courses]] * [https://www.filmmaking.net/film-schools Film Schools Directory] at filmmaking.net - a very large database of global film schools * Learn Filmmaking : [http://www.bhushanmahadani.com Learn Filmmaking Online] [[Category:Film School]] 3uisy81ceemvdcj9zyrvmhmnwvvmfv9 2691724 2691682 2024-12-13T00:15:25Z Atcovi 276019 Reverted edits by [[Special:Contributions/Dan Lock Author VR|Dan Lock Author VR]] ([[User_talk:Dan Lock Author VR|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2314862 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {{complete|text=This page is [[:Category:Completed resources|ready]] for students}}<br> {{Film School:Message|Style=Plain|Message=Enter the magical world of filmmaking.}} {| cellpadding="10" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: LemonChiffon; border: 1px solid Gray; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" | {{Courses in Filmmaking - Table - Float Right}} {{center top}}<br> {| border=0 cellspacing=0 cellpadding=5 | bgcolor="LemonChiffon" | [[Image:Crystal Clear app kfm home.png]] | bgcolor="LemonChiffon" | {{center top}}[[Film School/About|Wikiversity Film School]]<br><big>The Courses in<br>Narrative Film Production</big>{{center bottom}} |} {{center bottom}} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">[[Image:Crystal Clear action build.png|24px]] Click on the course name to begin the course. →</p> |} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">Note: These courses are only for dramatic motion pictures (movies with a script and dialog).<br>These courses are '''not''' for documentaries, event video, corporate video, educational programs, or multimedia.</p> {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 55%; background-color: Honeydew; border: 1px solid Gray; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="1" | [[Image:Nuvola apps edu miscellaneous.svg|right]] ===Anyone can learn to be a filmmaker=== ;Filmmaking is easy :Filmmaking is not rocket science. Everything about filmmaking is extremely easy to learn. Anyone can do it if they wish. ;The challenge of learning filmmaking :The challenge is filmmaking requires learning a huge number of skills. Each skill is easy to learn but the number of things you must learn is huge. :If you want to be an independent filmmaker, you must learn the equivalent of 20 different careers. Even if you are a fast learner, it can take you years to learn everything. ;Telling a story :In a dramatic motion picture, the story is told by many people. The [[wikipedia:Cinematographer|cinematographer]] tells the story with the camera. The lighting person tells the story with lighting. The film composer tells the story with music. The [[wikipedia:Actor|actors]] tell the story with action and dialog. The [[wikipedia:Film_editing|editor]] tells the story with editing. The [[wikipedia:Sound_design|sound designer]] tells the story with sound. ;You have to learn all of this :And as an independent filmmaker, you must learn all of these skills. :If you fail to learn even one of these skills, people will notice and be turned off by your movie. You must learn everything!!!! | style="width: 50%; background-color: #eef; border: 1px solid #777777; <!--vertical-align: top; -->-moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"; rowspan="3"| {{center top}}[[Image:Crystal_Clear_app_camera.png]]{{center bottom}} ==Film School Preparatory== :Wikiversity Film School is a {{font|color=Purple|'''preparatory school'''}} for budding filmmakers who plan to go to film school or take classes in motion picture production. :Each year, USC Film School receives 800 applicants to fill just 50 undergraduate positions. :We teach you the things that film schools expect you to know '''before''' you get to film school so that you can have a base to work from when you first start film-school. ==Learning filmmaking software== ;Intel processor :First, you need to do is learn about the computer software for filmmaking. See below for a list of the software you will need. Most is free! ;Designed for the Macintosh : The Film Scoring course is designed for Apple's GarageBand with Jam Pack<nowiki>:</nowiki> Symphony Orchestra :The Film Editing course is designed for Apple's Final Cut Pro or Adobe Premiere. ;Linux :Where possible, free Linux software is listed. :However, there is still no standard movie file format for Linux and nothing which matches the quality and usefulness of Apple's GarageBand for film scoring. :Therefore, Linux is currently not a good choice for filmmakers. |- | style="width: 50%; background-color: #fee; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"| [[Image:Nuvola apps kword.png|right|96px]] ==What do we do?== ;Make a tiny movie :In the filmmaking class, we create a short motion picture. It is less than a minute long so it is very simple. :But before you complete the animatic for the movie, you will also need to learn film [[Film editing|'''editing''']] and film [[Mad Max's Course in Film Scoring|'''scoring''']]. |- | style="width: 50%; background-color: #ffa; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"| [[Image:3D Universe Percy w cloths 1.png|right|90px]] ==Where to start?== ;Filmmaking, editing, and scoring :Budding filmmakers should start with the basic [[Filmmaking Basics|'''filmmaking course''']]. For the [[Filmmaking Basics/Formatting the Script|'''first lesson''']], you will format a very short motion picture script. After that, take the [[Filmmaking Basics/Thumbnail Pop Quiz|pop quiz]] where you tell me how you would begin to film this scene. :If you want to be a film composer (but you are not a musician), screen writer, or a film editor, you can go directly to the courses in [[Film editing|film editing]] or [[Mad Max's Course in Film Scoring|film scoring]] or the [[Filmmaking for High School Drama Departments|script writing exercise]]. :If you are a musician who wants to be a film composer, [[Film scoring/Introduction|click here]]. |} <p style = "font-size: 9pt;font-family: georgia; text-align: center; color: MidnightBlue">Note: These courses are designed for the Macintosh computer.<!--<br>PC World Magazines says, "The fastest Windows Vista notebook we've tested this year (through 10/25/07) is a Mac.<br>At 6.6 pounds and just 1 inch thick, the MacBook Pro is the lightest 17-inch notebook available."<br>Your instructor says, "Starting in 2008, if you are going to buy a new Windows PC, you might as well get a Macintosh.--> </p> {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 50%; background-color: #eef; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="1" | [[Image:Books-aj.svg_aj_ashton_01g.png|right|96px]] ==Cost== ;Lessons :These filmmaking courses are '''free'''. The only requirement is you submit your homework assignments so others can benefit. (see right) ;Software :I always try to select programs which are {{font|color=Green|'''free'''}}. | style="width: 50%; background-color: Honeydew; border: 1px solid #777777; <!--vertical-align: top; -->-moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="1"; rowspan="2"| <!--{{center top}}[[Image:Nuvola_apps_package_wordprocessing.png|96px]]{{center bottom}}--> [[Image:Nuvola_apps_package_wordprocessing.png|right|96px]] ==Sharing your work== ;Your homework assignments :Your completed assignments must be submitted under the Free Documentation License or as Public Domain. :This allows what you have learned to be shared by others. |} ---- ---- <p style = "font-size: 12pt;font-family: georgia; text-align: center; color: MidnightBlue">Free Software used at Wikiversity Film School</p> ---- ---- {| cellpadding="15" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 100%; background-color: LemonChiffon; border: 2px solid Blue; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:Crystal Clear app package.png|right|128px]] =={{font|color=Green|'''Free'''}} Filmmaking Software at Wikiversity Film School== :There is some of free software used at Wikiversity Film School. Sometimes, this software is demo versions or simplified versions of software but this is good enough to complete these lessons. Using {{font|color=Green|'''free'''}} software, you can learn a tremendous amount about filmmaking. |- | style="width: 100%; background-color: #ffe; border: 2px solid Blue; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:FrameForge SBTDS Screen Shot.png|right|260px]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">FrameForge 3D Studio Free Demo Version</p> ;Basic filmmaking (pre-production) :The heart of the Wikiversity Film School's basic filmmaking course (pre-production) is [http://www.frameforge3d.com/download.php '''FrameForge 3D Studio 2 Demo Version''']. This is the most educational program used at Wikiversity Film School. And the demo version is {{font|color=Green|'''free'''}} for both the Macintosh and Windows operating system. :This program simulates the motion picture camera, the movie set, and the actors. The program is useful for experimenting with the different lenses on your motion picture camera. And the final output of FrameForge 3D Studio 2 Demo Version is completely frame accurate (including depth of field effects) ready to give your cinematographer for creating all the shots of your movie. :The {{font|color=Green|'''free'''}} demo version is limited to 20 uses. Other limitations apply. However, this is more than enough to complete your assignments at Wikiversity Film School. Download this '''free''' program today and begin learning how to use it. ---- [[Image:GarageBand Musical Typing.jpg|260px|right]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">Apple's GarageBand</p> ;Film Scoring :All film scoring lessons at Wikiversity Film School can be done using Apple's GarageBand which is {{font|color=Green|'''free'''}} with each new Macintosh PC. :In addition to a midi program like GarageBand, you will also need a good selection of musical instruments for the symphony orchestra. Some musical software instruments of the symphony orchestra for GarageBand are {{font|color=Green|'''free'''}} (such as the packages from Boldt) and some are not. I recommend Apple's Jam Pack:Symphony Orchestra. I have not tried Apple's new Jam Pack for voices which should also be useful. :*Filmmakers should {{font|color=OrangeRed|'''NOT'''}} use Apple's Logic 7 which is poorly designed and exceedingly awkward. I have not tried version 8 yet. :* I do not know what to recommend for Windows. ---- [[Image:The Assignment Screen Shot.png|right|140px]] <p style = "font-size: 16pt;font-family: georgia; text-align: center; color: MidnightBlue">Film Dailies</p> ---- ;Animation :Starting in 2009, Wikiversity Film School will have a simple course in 3D animation. This will use the {{font|color=Green|'''free'''}} programs of DAZ Studio and Bryce 5.5. : If you have Poser and Vue, you can use those programs but they are not free. :DAZ Studio and Bryce take time to learn so if you are interested in working with 3D, you should download these {{font|color=Green|'''free'''}} programs and begin to learn them. ---- [[Image:ArtRage2Chalk100.jpg|right|200px]] ;Other useful programs :*[http://www.ambientdesign.com/artragedown.html ArtRage 2.5 Starter Edition] is a {{font|color=Green|free}} paint program used for matte painting and creating the movie poster. This is a simple and fun artistic painting program. Feels very natural. Extremely useful. If you need any kind of digital painting with traditional artist materials, this is your first choice. This is the {{font|color=Green|'''free'''}} version which is fully working but more limited than the full version which is also remarkably inexpensive. :* [https://gimp.en.softonic.com/mac GIMP] is a {{font|color=Green|'''free'''}} paint program comparable to Adobe Photoshop. Surprisingly mature program but definitely not as easy to use as Photoshop. Requires X-11. If you do not have Adobe Photoshop 4 or later, this program is absolutely necessary for some of the classes at Wikiversity Film School. :*[http://www.audacityteam.org/download/ Audacity] is a {{font|color=Green|'''free'''}} useful utility for working with audio and converting audio to OGG files. It is a bit awkward but it is free. :* [http://www.tuxpaint.org/ Tux Paint] is a {{font|color=Green|'''free'''}} fun paint program for kids. It has the added advantage of using rubber stamps and creating matte paintings easily. Has the unique feature of doing rough storyboards with special storyboard artwork. Also useful for one of the lessons about learning matte painting at Wikiversity Film School for kids. |} {| cellpadding="25" cellspacing="5" style="width: 100%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 50%; background-color: #fed; border: 1px solid #777777; vertical-align: bottom; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="2" | [[Image:Crystal Clear app korganizer.png|128px|right]] ==Where to begin?== :*If you want to become a filmmaker, start with the '''[[Filmmaking Basics|Basic Filmmaking Course]]'''. [[Image:Mplayer.svg|32px]] :*Later, when you are ready, you should also take the '''[[Film editing|Film Editing Course]]'''. [[Image:Crystal Clear mimetype video.png|32px]] :*The most important course of all is film scoring. You will need the '''[[Film scoring| Film Scoring Course]]''' to complete the other two courses. [[Image:Crystal Clear app knotify.png|32px]] {{Contact your instructor: Filmmaking|page={{PAGENAME}}}} |} <!-- {{center top}}<small>[[Film school:site map|site map]] · [[Film school:Completed Homework Assignments|homework]]</small>{{center bottom}} --> ==External links== ;{{font|color=Red|NOT}} RECOMMENDED for these lessons :As far as I know, currently none of the programs listed below is compatible with the disks used in these lessons. -- ''Robert Elliott, your Instructor'' :* [http://www.musix.org.ar/en/index.html Musix] - A multimedia Linux distro (music and video) :* [http://jahshaka.org/ Jashaka] - comprehensive open source player and editor - "Powering the new Hollywood" :* [http://www.virtualdub.org/ VirtualDub] - Open source nonlinear editor with recent release :* [http://directory.fsf.org/category/vmanip/ FSF] - list of free video editing tools :* [http://fixounet.free.fr/avidemux/ Avidemux] - open source video editor * [http://digitaltippingpoint.com/wiki/index.php?title=Main_Page Digital Tipping Point wiki] - Collaborative open documentary ---- ---- * <small>Student in the Basic Filmmaking Course may [[Film school - Submitting student assignments - Basic filmmaking course|upload their own assignments]].</small> * <small>Student in the Film Scoring Course may [[Film school - Submitting student assignments - Film scoring course|upload their own assignments]].</small> * <small>Student in the Film Editing Course may [[Film school - Submitting student assignments - Film editing course|upload their own assignments]].</small> [[Category:{{PAGENAME}}| ]] [[Category:Narrative Film Editing]] [[Category:Filmmaking in the Drama Department]] [[Category:Film scoring]] [[Category:Courses]] * [https://www.filmmaking.net/film-schools Film Schools Directory] at filmmaking.net - a very large database of global film schools * Learn Filmmaking : [http://www.bhushanmahadani.com Learn Filmmaking Online] [[Category:Film School]] n11rwbsyc255xsxsivfiu5qrbojyflq 0 28306 2691774 2611505 2024-12-13T09:53:31Z 117.199.128.193 NOTHING 2691774 wikitext text/x-wiki {{TOCright}} '''Research''' is defined as human activity based on intellectual application in the investigation of matter. The primary purpose for applied research is discovering, interpreting, and the development of methods and systems for the advancement of human knowledge on a wide variety of scientific matters of our world and the universe. Research can use the scientific method, but need not do so. ==Research resources== * [[Wikiversity: Research ethics]] * [[Presentations]] - * [[High risk research]] - High risk research can be discussed at that location. * [[Business research]] [[Category:Presentations]] * [[Qualitative research]] ==Category trees== <div style="column-count:3;-moz-column-count:3;-webkit-column-count:3">{{#categorytree:{{PAGENAME}}|hideroot|mode=pages}}</div> ==See also== * [[Portal:Research]] - user-friendly guide to the [[:Category:Research|Research category]] * [[Wikiversity:Research]] - research policy * [[Wikiversity:Research process]] * [[:Category:Research policy proposals|Research policy proposals]] * [[Wikiversity:Publishing original research|Publishing original research]] * [[Data Analysis using the SAS Language]] * [[Research and Engagement Funding in the UK]] * [[Online surveys]] ==External links == * [http://sciencenow.sciencemag.org/cgi/content/full/2007/307/1?rss=1 Pro-research patent reform at universities.] * [http://pkp.sfu.ca/ Public Knowledge Project] * [http://highwire.stanford.edu/lists/freeart.dtl Highwire Press] - 1,776,700 free full-text articles :*[http://highwire.stanford.edu/lists/largest.dtl Other free full text archives] * [http://www.leedsmet.ac.uk/lskills/local/sfl2/content/research Research skills] (Leeds Metropolitan University) * [http://www.unu.edu/Unupress/fulltext.html Some full text publications] ===Medical research=== * [http://nihroadmap.nih.gov/ NIH Roadmap for Medical Research] [[Category:Research| ]] [[ru:Научные исследования]] fktl0jj3wvotybyxxdxn2vbxj3ol0co 0 34668 2691642 2687745 2024-12-12T15:28:40Z Lbeaumont 278565 /* Tips for better writing (mostly for English) */ Added dangling modifiers 2691642 wikitext text/x-wiki This lesson provides a comprehensive exploration of the art of effective writing, providing a universal introduction suitable for all ages. [[File:Fountain pen writing (literacy).jpg|thumb|Good writing is clear thinking made visible.]] {{TOC right | limit|limit=2}} '''Writing''', a uniquely human tool, serves as a visible expression of language, allowing the conveyance of thoughts in diverse ways. Much like spoken language, the possibilities for arranging words are infinite, enabling the communication of ideas, thoughts, images, and emotions. Whether the aim is to inform, persuade, entertain, or a combination of these, the written word remains a powerful means of expression. In the realm of written language, several fundamental components shape the structure of writing. Paragraphs, separated by indentations or line breaks, serve as building blocks. Words, composed of letters, are the basic units of expression. Punctuation marks guide the reader, providing cues for timing and emphasis. Sentences function to encapsulate concise ideas, while paragraphs unite sentences to convey a broader concept. [[Essay|Essays]], whether lengthy or concise, compile paragraphs into cohesive compositions. The diversity of essays spans various types, each serving unique purposes. In the realm of [[fiction]], ideas take on imaginative forms, crafted by the writer's creativity. Conversely, non-fiction endeavors to explore ideas and concepts grounded in reality. This lesson invites writers of all ages to delve into the intricacies of written expression, fostering an understanding of the myriad ways in which words can shape, inform, and enrich our communication. == Tips for better writing (mostly for English) == *Overcome [[/writers block/|writer's block]]. * Follow the [[/basic grammar rules of American English writing/]]. * If you are required to follow a specific [[w:Style_guide|style guide]] or style manual, then: *# obtain a copy of that guide *# study it, and *# follow the requirements of that guide. * Use [[Writing/punctuation marks|correct punctuation]]. ** Full stops, also know as the [[w:Full_stop|period]] character, (.) separate [[w:Sentence_(linguistics)#By_purpose|declarative]] sentences. The [[w:Question_mark|question mark]] (?) is used at the end of a question, and an [[w:Exclamation_mark|exclamation point]] (!) is used after an [[w:Interjection|interjection]] or [[w:Sentence_(linguistics)|exclamation]] to indicate strong feelings or to show [[w:emphasis|emphasis]]. ** Know how to use ''it's'' and ''its''. ''It's'' is always a contraction of "it is"; ''its'' is possessive ("belonging to it"). * [[Writing/Correct use of capital letters|Use capital letters correctly]]. * [[/Understanding and Fixing Dangling Modifiers/|Avoid dangling modifiers.]] * Use the [[/paragraph as the major organizing element/]]. * Choose [[/suitable sentence length/]]. * [[Good Writing is Clear Thinking Made Visible|Good writing is clear thinking made visible]]. * Be [[Intellectual honesty|Intellectually honest]]. ** [[Living Wisely/Advance no falsehoods|Advance no falsehoods]]. * [[/Good Writing is Precise and Concise/]]. * Choose [[Writing/precise descriptive, and engaging language|precise descriptive, and engaging language]]. ** Do not use words like "very", "good", "get", "thing", or "things" if it can be avoided. * Choose to use or avoid [[/contractions/]] based on the context and [[Writing/Tone|tone]] of the writing. * Do not confuse words (known as [[w:Homonym|homonyms]]) that sound the same yet are spelled differently. ** Examples include: ''their'', ''they're'' and ''there''; ''weather'' and ''whether''; etc. ** Refer to this longer list of [[Writing/commonly_confused_homonyms|commonly confused homonyms]] to avoid misuse. * [[/Read extensively/]] to refine your craft and continually evolve. * Use [[Writing/transition phrases|transition phrases]] skillfully. * Favor [[/active voice/]] over passive voice. * Use [[/poetic phrases/]] skillfully. * Whenever possible, [[/show rather than tell/]] to describe the scene. * Use a [[/variety of phrases/]] during dialogue. * Strive to create [[/great writing/]]. ** [[/Great writing is clever writing/]]. ** [[/Great writing is witty/]]. * [[/Choose the title/]] carefully. It is often best to keep the title concise and to the point. * Write [[Candor|candidly]]. Express your opinions clearly, accurately, and [[Finding Courage|courageously]]. * [[/Requesting Feedback/|Request feedback]]. Improve the work based on the feedback received. * Request writing [[/assistance from ChatGPT/]] (or other [[w:Large_language_model|LLM]]). ** Acknowedge any assistance you receive. * Consider this [[/hypothetical advice on writing from great thinkers/]]. * Edit and [[w:Proofreading|proofread]] everything, even if it is a one-page essay or short [[Email Checklist|email]]. ==Collaborative papers== You can start a collaborative paper in the main namespace, or if you would like to create something individually, please keep it in the user namespace on your userpage or as a subpage to your userpage. * [[Perfection]] * [[On Hatred]] * [[Privacy, Security, and Implied Mutual Exclusion]] ==See also== * [[Writing arts|Writing Arts]] * [[Academic and Legal Research and Writing]] * [[Writing prompts]] ==External links== * [http://www.crockford.com/wrrrld/style.html The Elements of Style by William Strunk, Jr.] [[Category:Writing]] fgqw492pv6bl07ehq56y0ylswbd3izc 0 52529 2691617 1419612 2024-12-12T13:18:18Z 27.147.228.7 Reference 2691617 wikitext text/x-wiki ==Challenges with using technology in schools== * School technology systems are frequently broadcast structured. * Read/write internet spaces are often blocked. It can be difficult for teachers to use resources which may then be blocked. * Bandwidth and technology purchases are often delivered from a centralised source. Despite this bandwidth costs can be high in schools. Costs of uploading can be passed on to specific students/parents and connections can be for restricted lengths of time. The choices are made centrally but costs are experienced locally. * There is little room for teacher or student responsibility or ability to pull value for specific learning at the fingertip end of the system. * It has been difficult to secure support staff in schools and to find opportunities for professional development. * Buildings may not be designed for adaptive work with technology. * Timetables may not be structured for experimentation. * Assessments are ultimately geared for standardised testing. * What kinds of assessment encourage and record excellence in constructivist education and student leadership. ==Opportunities with free software== Some aspects of the openness of free software may help directly with these issues. * If technology in schools can be supported in ways which enable local choice, experimentation, and constructivist or connectivist pedagogy would fit better. eg. It is legal to 'look under the bonnet' of open source software. * If students and teachers are able to make their own choices and experiments with choosing technologies they can be more adaptive. Some schools do these things now. * Offline storage of software and resources. Software can be taken home and shared with friends legally. * burning runs of OpenCD(OpenDisc) software * recycling computers as a community service and learning project * building networks with student participation * Education systems development could be a part of the shared responsibility and development process which applies to other aspects of education. * Teachers, curriculum, students and increasingly parents and other education providers are potential partners in these systems. * Software which complies to open standards makes it possible to share information between a range of tools. * Open source tools can be adapted to suit specific requirements. ==Choosing technology for learning== ===Curiosity=== Stallman suggests open source technology offers particular opportunities in education contexts. <blockquote>Free software permits students to learn how software works. When students reach their teens, some of them want to learn everything there is to know about their computer system and its software. That is the age when people who will be good programmers should learn it. To learn to write software well, students need to read a lot of code and write a lot of code. They need to read and understand real programs that people really use. They will be intensely curious to read the source code of the programs that they use every day. Proprietary software rejects their thirst for knowledge: it says, “The knowledge you want is a secret—learning is forbidden!” Free software encourages everyone to learn.</blockquote> Hall J has also written and spoken about the way that open code adds dimension to tools in education. <blockquote>Free Software teaches you twice; once when you use it as a tool, and once when you view the source code to see how the tool works.</blockquote> Open licences permit software users to look under the bonnet, to find out how things work: Their curiosity is not scoped by concerns about whether it is legal to know how something works. Open source software can be used in contexts where the learning is student led because there is no question they may not ask. As a result FOSS people engage with information and technology as makers and participants rather than consumers. This difference in relationship creates opportunities for teachng and learning. Students and teachers are able to choose between using a tool or engaging with the tool and its community as participants in the project community. Free access to the tools for learning beyond educational institutions encourages learners to see that learning at school is a starting point. The ideal is that homework is not set but students want to do 'homework' because they are inspired and have a need to master. FOSS provides them with the freedom to take their learning further and explore and customise the tools they use. ===Multiple perspectives=== Tools like wikis can be structured to enable parallel truths and negotiated outcomes. Some topics in the campaigns wikia were structured this way to try and explore the wiki as an expression of contrasting perspectives around an idea. An example of the structure (but not the process at this stage) is at the Digital Rights topic. http://campaigns.wikia.com/wiki/Digital_Rights ===Tools that grow with you=== The olpc project has been working towards a fully open source laptop. Their reasons for choosing open source solutions are described by Mako Hill <blockquote>The Laptop will bring children technology as means to freedom and empowerment. The success of the project in the face of overwhelming global diversity will only be possible by embracing openness and by providing the laptop's users and developers a profound level of freedom. As the children grow and pursue new ideas, the software and the tools should be able to grow with them and provide a gateway to other technology. </blockquote> ===Technology for innovation === In a classroom context a community of students engaging in contention and negotiation in diverse activities without precedent or finite scope is less system friendly. It does offer opportunities for students to learn by trying and to be inquisitive and experiment freely. Making technical and social mistakes is possible. Learning from them is also possible. The systems which support and govern a classroom engaged in experimental or participative practice may need to think differently about kinds of space, time and ways of valuing which scope the opportunity if they are to keep the flexibility which makes it possible for students to be innovative. If there are different needs in a class context, there are also different opportunities when the student is at home. If open software used and the technology projects or communities are open online it is possible for them to participate when they are at home, in hospital, or relocating interstate. This poses different questions about access which relate more to whether students have access to facilities outside of school which make open participation possible. Again our education systems are more closely tied to the industrial entity of school and do not generally think of learning as something which a student might need independent access to regardless of institutional affiliation. These questions all pivot around being able to see what is valuable from the student's perspective and what is valuable from the wider community perspective and having some connection between those two conversations. OReilly 2000 describes the negotiation and innovation which are a part of the challenge and reward of open source participation: <blockquote>I'd like to argue that open source is the "natural language" of a networked community, that the growth of the Internet and the growth of open source are interconnected by more than happenstance. As individuals found ways to communicate through highly leveraged network channels, they were able to share information at a new pace and a new level. Just as the spread of literacy in the late middle ages disenfranchised old power structures and led to the flowering of the Renaissance, it's been the ability of individuals to share knowledge outside the normal channels that has led to our current explosion of innovation. Just as ease of travel helped new ideas to spread, wide area networking has allowed ideas to spread and take root in new ways. Open source is ultimately about communication.</blockquote> ==Technology support== ===System wide tools=== There may be some aspects of school technologies which need to be systemic in approach. There are many good free open source technologies which are built on the system wide model. Many of the underlying internet technologies are open source. Free open source software customisation can occur in Australia, in a state or to suit the needs of a school or region. Open licences <ref>{{Cite web|url=https://www.icslegal.co/|title=Apply for a Sponsor Licence {{!}} Step-by-Step Guidance UK & US|website=Sponsor Management System|language=en|access-date=2024-12-12}}</ref> ensure the technology will always be accessible and editable. ===Saving bandwidth and finding local value=== Bandwidth costs are often a scoping factor for schools. Internet costs are often experienced in a classroom context as an intermittent connection, limited access to some kinds of services eg video or sound, or as a pay to play cost for the school or for parents. Australia's decision to manage internet access in this way in a school context has a very strong message that students are consumers who must pay to access centrally held knowledge. Australia's high bandwidth costs are based around traffic over a connection. Other nations structure their connections around the provision of connectivity as the traffic component is an artificial cost. It would be useful to review bandwidth cost models and to assess whether there may be better ways to provide good connectivity. Perhaps schools could peer or mirror information to reduce bandwidth costs. School networks could share information between schools directly as peer nodes on a network to reduce bandwidth costs. This kind of local WAN information sharing could also encourage opportunities to promote local content and collaboration. Students and teachers are the primary creators of information directly related to their curriculum. Our ability to see value in our own Australian participation could be reflected in the ways that we structure these networks for student and teacher content as well as for connections with wider communities of interest. For example a teacher may use a podcast to enable students to call in with questions around a biology program. The questions could relate to the course or to local community experience. Other students might call and ask too. That podcast might be useful in other schools. How can those kinds of connections be effectively sponsored in the wider system. Taking on board the implications of distribtued publishing provides very different opportunities and also suggests that we should choose to reconsider the ways that we structure internet connectivity both in a logical routing sense and as a financial tollway on learning. ===Local choice=== It is also important to look for ways to support choice in the classroom wherever possible because the 'room to move' provided to teachers and students is important for initiative and creativity and responding to individual opportunities and challenges. ===Building support capacity with student partners=== Local adaptation and maintenance are important for flexibility in the school context. Students/cadets studying part time in VET or highered could be employed as trainee technical support on school networks. Mentoring from VET/highered courses and from internal mentors as well as from external open communities would be a nice way for students to find their feet in networking technologies. As with many of the other student led processes there would need to be a different overall philosophy operating around the network. A section of network which is not critical and which is able to be used as a testing ground would be useful. Sometimes breaking things might be a part of the process of making the network and the students more useful/skilled. ===Standards based technologies=== Real open standards for protocols and document formats are frequently a priority in FOSS projects because the open standards are a meeting point for the diverse community of different tools. These standards also make it possible for software to be integrated with other solutions and for the school communities to have data security into the future. Choosing tools which use open standards so that data can be imported and exported makes the choice of tool a softer decision because the data will be accessible regardless of the prior choices made. ===Working offline=== Some kinds of information would be more accessible on local cd drives or usb sticks for take home use. Making sure that materials and tools are able to e used in situations where students or schools may not have a working connection provides better resilience. === Free and open source software fostering careers === Use of FOSS in education sectors means that; * the projects can become opportunities for further community participation, education, research or employment. * skills in negotiation around project criteria are vital in innovative practice where there may not be an existing correct answer. * long term participation in FOSS projects provides a public record of dialogue and contributions which technology companies are increasingly using as indicators of skill and commitment when looking for new hires. ==Resources== [[FOSSlike_collaboration_in_schools/Resources|Resources]] 5hm24a1izqtgi7dhq98dv413h2utf2l 0 54688 2691723 2677525 2024-12-12T23:49:28Z Huz and Buz 2928717 Undo revision [[Special:Diff/2677525|2677525]] by [[Special:Contributions/2A00:A041:72A0:1100:510D:42D:7DB3:4C2D|2A00:A041:72A0:1100:510D:42D:7DB3:4C2D]] ([[User talk:2A00:A041:72A0:1100:510D:42D:7DB3:4C2D|talk]]) The page is based on the KJV, which is imperfect but its open source, it should be kept consistent. 2691723 wikitext text/x-wiki According to the [[w:documentary hypothesis|documentary hypothesis]], [[w:Genesis|Genesis]] is composed from a number of originally independent sources joined by a [[w:redaction|redactor]]. There follows the text of Genesis in the [[w:King James Version|King James Version]], with sources highlighted according to the documentary hypothesis. Further subdivisions of the main sources are viewable by reading the individual source pages. *The [[w:Priestly source|"Priestly source"]] is highlighted in {{font|color=#888800|olive yellow}} <small>{{font|color=#0000FF|([[Bible/King James/Documentary Hypothesis/Priestly source|view in isolation]])}}</small> *The [[w:Jahwist|"Jahwist source"]] is highlighted in {{font|color=#000088|navy blue}} <small>{{font|color=#0000FF|([[Bible/King James/Documentary Hypothesis/Jahwist source|view in isolation]])}}</small> *The [[w:Elohist|"Elohist source"]] is highlighted in {{font|color=#008888|teal blueish grey}} <small>{{font|color=#0000FF|([[Bible/King James/Documentary Hypothesis/Elohist source|view in isolation]])}}</small> *The [[w:Torah redactor|"Additions by the Redactor and other late insertions"]] are highlighted in {{font|color=#880000|maroon red}} ==Chapter 1== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| In the beginning God created the heaven and the earth.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1278316552606158848|title=Genesis 1:1|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| And the earth was without form, and void; and darkness upon the face of the deep. And the Spirit of God moved upon the face of the waters.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1278658802162839552|title=Genesis 1:2|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And God said, Let there be light: and there was light.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And God saw the light, that it was good: and God divided the light from the darkness.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And God called the light Day, and the darkness he called Night. And the evening and the morning were the first day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1279015701412483072|title=Genesis 1:3-5|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And God made the firmament, and divided the waters which under the firmament from the waters which above the firmament: and it was so.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And God called the firmament Heaven. And the evening and the morning were the second day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1279418701452804096|title=Genesis 1:6-8|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And God said, Let the waters under the heaven be gathered together unto one place, and let the dry appear: and it was so.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And God called the dry Earth; and the gathering together of the waters called he Seas: and God saw that it was good.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And God said, Let the earth bring forth grass, the herb yielding seed, and the fruit tree yielding fruit after his kind, whose seed is in itself, upon the earth: and it was so.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And the earth brought forth grass, and herb yielding seed after his kind, and the tree yielding fruit, whose seed was in itself, after his kind: and God saw that it was good.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And the evening and the morning were the third day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1279742118097977350|title=Genesis 1:9-13|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And God said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days, and years:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And let them be for lights in the firmament of the heaven to give light upon the earth: and it was so.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And God made two great lights; the greater light to rule the day, and the lesser light to rule the night: he made the stars also.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And God set them in the firmament of the heaven to give light upon the earth,}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And to rule over the day and over the night, and to divide the light from the darkness: and God saw that it was good.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And the evening and the morning were the fourth day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1280106889414615047|title=Genesis 1:14-19|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And God said, Let the waters bring forth abundantly the moving creature that hath life, and fowl that may fly above the earth in the open firmament of heaven.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| And God blessed them, saying, Be fruitful, and multiply, and fill the waters in the seas, and let fowl multiply in the earth.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And the evening and the morning were the fifth day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1280461026484502528|title=Genesis 1:20-23|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And God said, Let the earth bring forth the living creature after his kind, cattle, and creeping thing, and beast of the earth after his kind: and it was so.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| And God made the beast of the earth after his kind, and cattle after their kind, and every thing that creepeth upon the earth after his kind: and God saw that it was good.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| So God created man in his own image, in the image of God created he him; male and female created he them.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#888800| And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#888800| And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#888800| And God saw every thing that he had made, and, behold, it was very good. And the evening and the morning were the sixth day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1280828033281253376|title=Genesis 1:24-31|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> ==Chapter 2== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| Thus the heavens and the earth were finished, and all the host of them.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| These are the generations of the heavens and of the earth when they were created,}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1281192949754273796|title=Genesis 2:1-4a|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|color=#000088| in the day that the LORD}} {{font|color=#880000| God}}<ref name=":0">{{Cite web|url=https://twitter.com/joelbaden/status/1281559839760691200|title='LORD God'|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|color=#000088| made the earth and the heavens,}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And every plant of the field before it was in the earth, and every herb of the field before it grew: for the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| had not caused it to rain upon the earth, and there was not a man to till the ground.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| But there went up a mist from the earth, and watered the whole face of the ground.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1281559834983358464|title=Genesis 2:4b-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-05}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| planted a garden eastward in Eden; and there he put the man whom he had formed.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And out of the ground made the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of knowledge of good and evil.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And a river went out of Eden to water the garden; and from thence it was parted, and became into four heads.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| The name of the first is Pison: that is it which compasseth the whole land of Havilah, where there is gold;}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And the gold of that land is good: there is bdellium and the onyx stone.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And the name of the second river is Gihon: the same is it that compasseth the whole land of Ethiopia.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And the name of the third river is Hiddekel: that is it which goeth toward the east of Assyria. And the fourth river is Euphrates.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| took the man, and put him into the garden of Eden to dress it and to keep it.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| commanded the man, saying, Of every tree of the garden thou mayest freely eat:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1281914563735298048|title=Genesis 2:8-17|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| said, It is not good that the man should be alone; I will make him an help meet for him.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And out of the ground the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field; but for Adam there was not found an help meet for him.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof;}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And the rib, which the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| had taken from man, made he a woman, and brought her unto the man.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And they were both naked, the man and his wife, and were not ashamed.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1282271376972316672|title=Genesis 2:18-25|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> ==Chapter 3== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Now the serpent was more subtil than any beast of the field which the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| had made. And he said unto the woman, Yea, hath God said, Ye shall not eat of every tree of the garden?}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And the woman said unto the serpent, We may eat of the fruit of the trees of the garden:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| But of the fruit of the tree which is in the midst of the garden, God hath said, Ye shall not eat of it, neither shall ye touch it, lest ye die.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And the serpent said unto the woman, Ye shall not surely die:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And when the woman saw that the tree was good for food, and that it was pleasant to the eyes, and a tree to be desired to make one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the eyes of them both were opened, and they knew that they were naked; and they sewed fig leaves together, and made themselves aprons.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1282631958271582209|title=Genesis 3:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And they heard the voice of the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| walking in the garden in the cool of the day: and Adam and his wife hid themselves from the presence of the LORD }} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| amongst the trees of the garden.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| called unto Adam, and said unto him, Where art thou?}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he said, I heard thy voice in the garden, and I was afraid, because I was naked; and I hid myself.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And he said, Who told thee that thou wast naked? Hast thou eaten of the tree, whereof I commanded thee that thou shouldest not eat?}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And the man said, The woman whom thou gavest to be with me, she gave me of the tree, and I did eat.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| said unto the woman, What is this that thou hast done? And the woman said, The serpent beguiled me, and I did eat.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| Unto the woman he said, I will greatly multiply thy sorrow and thy conception; in sorrow thou shalt bring forth children; and thy desire shall be to thy husband, and he shall rule over thee.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life;}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field;}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Adam called his wife's name Eve; because she was the mother of all living.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Unto Adam also and to his wife did the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| make coats of skins, and clothed them.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| Therefore the LORD}} {{font|color=#880000| God}}<ref name=":0" /> {{font|color=#000088| sent him forth from the garden of Eden, to till the ground from whence he was taken.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1283002016395153409|title=Genesis 3:8-24|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> ==Chapter 4== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Adam knew Eve his wife; and she conceived, and bare Cain, and said, I have gotten a man from the LORD.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And she again bare his brother Abel. And Abel was a keeper of sheep, but Cain was a tiller of the ground.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And in process of time it came to pass, that Cain brought of the fruit of the ground an offering unto the LORD.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And Abel, he also brought of the firstlings of his flock and of the fat thereof. And the LORD had respect unto Abel and to his offering:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| But unto Cain and to his offering he had not respect. And Cain was very wroth, and his countenance fell.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And the LORD said unto Cain, Why art thou wroth? and why is thy countenance fallen?}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| If thou doest well, shalt thou not be accepted? and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1283367662450442240|title=Genesis 4:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Cain talked with Abel his brother: and it came to pass, when they were in the field, that Cain rose up against Abel his brother, and slew him.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And the LORD said unto Cain, Where is Abel thy brother? And he said, I know not: Am I my brother's keeper?}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he said, What hast thou done? the voice of thy brother's blood crieth unto me from the ground.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And now art thou cursed from the earth, which hath opened her mouth to receive thy brother's blood from thy hand;}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| When thou tillest the ground, it shall not henceforth yield unto thee her strength; a fugitive and a vagabond shalt thou be in the earth.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And Cain said unto the LORD, My punishment is greater than I can bear.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Behold, thou hast driven me out this day from the face of the earth; and from thy face shall I be hid; and I shall be a fugitive and a vagabond in the earth; and it shall come to pass, that every one that findeth me shall slay me.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And the LORD said unto him, Therefore whosoever slayeth Cain, vengeance shall be taken on him sevenfold. And the LORD set a mark upon Cain, lest any finding him should kill him.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Cain went out from the presence of the LORD, and dwelt in the land of Nod, on the east of Eden.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1283736246058131457|title=Genesis 4:8-16|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And Cain knew his wife; and she conceived, and bare Enoch: and he builded a city, and called the name of the city, after the name of his son, Enoch.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And unto Enoch was born Irad: and Irad begat Mehujael: and Mehujael begat Methusael: and Methusael begat Lamech.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And Lamech took unto him two wives: the name of the one was Adah, and the name of the other Zillah.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Adah bare Jabal: he was the father of such as dwell in tents, and of such as have cattle.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And his brother's name was Jubal: he was the father of all such as handle the harp and organ.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Zillah, she also bare Tubalcain, an instructer of every artificer in brass and iron: and the sister of Tubalcain was Naamah.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Lamech said unto his wives, Adah and Zillah, Hear my voice; ye wives of Lamech, hearken unto my speech: for I have slain a man to my wounding, and a young man to my hurt.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| If Cain shall be avenged sevenfold, truly Lamech seventy and sevenfold.}}<ref name=":23">{{Cite book|url=https://www.worldcat.org/oclc/784958108|title=The composition of the Pentateuch : renewing the documentary hypothesis|last=Baden|first=Joel S.|date=2012|publisher=Yale University Press|isbn=978-0-300-15264-7|location=New Haven|oclc=784958108}}</ref> {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And Adam knew his wife again; and she bare a son, and called his name Seth: For God, said she, hath appointed me another seed instead of Abel, whom Cain slew.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And to Seth, to him also there was born a son; and he called his name Enos: then began men to call upon the name of the LORD.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1284100249464836097|title=Genesis 4:17-26|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> ==Chapter 5== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| This is the book of the generations of Adam. In the day that God created man, in the likeness of God made he him;}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| Male and female created he them; and blessed them, and called their name Adam, in the day when they were created.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Adam lived an hundred and thirty years, and begat a son in his own likeness, after his image; and called his name Seth:}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And the days of Adam after he had begotten Seth were eight hundred years: and he begat sons and daughters:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And all the days that Adam lived were nine hundred and thirty years: and he died.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And Seth lived an hundred and five years, and begat Enos:}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And Seth lived after he begat Enos eight hundred and seven years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And all the days of Seth were nine hundred and twelve years: and he died.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And Enos lived ninety years, and begat Cainan:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And Enos lived after he begat Cainan eight hundred and fifteen years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And all the days of Enos were nine hundred and five years: and he died.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And Cainan lived seventy years, and begat Mahalaleel:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And Cainan lived after he begat Mahalaleel eight hundred and forty years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And all the days of Cainan were nine hundred and ten years: and he died.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And Mahalaleel lived sixty and five years, and begat Jared:}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And Mahalaleel lived after he begat Jared eight hundred and thirty years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And all the days of Mahalaleel were eight hundred ninety and five years: and he died.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And Jared lived an hundred sixty and two years, and he begat Enoch:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And Jared lived after he begat Enoch eight hundred years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And all the days of Jared were nine hundred sixty and two years: and he died.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And Enoch lived sixty and five years, and begat Methuselah:}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| And Enoch walked with God after he begat Methuselah three hundred years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And all the days of Enoch were three hundred sixty and five years:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And Enoch walked with God: and he was not; for God took him.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| And Methuselah lived an hundred eighty and seven years, and begat Lamech:}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| And Methuselah lived after he begat Lamech seven hundred eighty and two years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| And all the days of Methuselah were nine hundred sixty and nine years: and he died.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And Lamech lived an hundred eighty and two years, and begat}}<ref name=":55">{{Cite web|url=https://twitter.com/joelbaden/status/1284457174727634946|title=Genesis 5|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|color=#880000| a son:}}<ref name=":55" /> {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And he called his name Noah, saying, This same shall comfort us concerning our work and toil of our hands, because of the ground which the LORD hath cursed.}}<ref name=":55" /> {{font|size=smaller|color=#0000FF|30}}{{font|color=#888800| And Lamech lived after he begat Noah five hundred ninety and five years, and begat sons and daughters:}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#888800| And all the days of Lamech were seven hundred seventy and seven years: and he died.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#888800| And Noah was five hundred years old: and Noah begat Shem, Ham, and Japheth.}}<ref name=":55" /> ==Chapter 6== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And it came to pass, when men began to multiply on the face of the earth, and daughters were born unto them,}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| That the sons of God saw the daughters of men that they were fair; and they took them wives of all which they chose.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And the LORD said, My spirit shall not always strive with man, for that he also is flesh: yet his days shall be an hundred and twenty years.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bear children to them, the same became mighty men which were of old, men of renown.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1284827282000027650|title=Genesis 6:1-4|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And God saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And it repented the LORD that he had made man on the earth, and it grieved him at his heart. }} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the LORD said, I will destroy man whom I have created from the face of the earth;}}<ref name=":1">{{Cite web|url=https://twitter.com/joelbaden/status/1285161918056079360|title=Genesis 6:5-13|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|color=#880000|both man, and beast, and the creeping thing, and the fowls of the air;}}<ref name=":1" /> {{font|color=#000088|for it repenteth me that I have made them. }} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| But Noah found grace in the eyes of the LORD.}}<ref name=":1" /> {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| These are the generations of Noah:}}<ref name=":1" /> {{font|color=#000088|Noah was a just man and perfect in his generations,}}<ref name=":1" /> {{font|color=#888800|and Noah walked with God.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And Noah begat three sons, Shem, Ham, and Japheth.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| The earth also was corrupt before God, and the earth was filled with violence.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And God looked upon the earth, and, behold, it was corrupt; for all flesh had corrupted his way upon the earth.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And God said unto Noah, The end of all flesh is come before me; for the earth is filled with violence through them; and, behold, I will destroy them with the earth.}}<ref name=":1" /> {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| Make thee an ark of gopher wood; rooms shalt thou make in the ark, and shalt pitch it within and without with pitch.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And this is the fashion which thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| A window shalt thou make to the ark, and in a cubit shalt thou finish it above; and the door of the ark shalt thou set in the side thereof; with lower, second, and third stories shalt thou make it.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And, behold, I, even I, do bring a flood of waters upon the earth, to destroy all flesh, wherein is the breath of life, from under heaven; and every thing that is in the earth shall die.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| But with thee will I establish my covenant; and thou shalt come into the ark, thou, and thy sons, and thy wife, and thy sons' wives with thee.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And of every living thing of all flesh, two of every sort shalt thou bring into the ark, to keep them alive with thee; they shall be male and female.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| Of fowls after their kind, and of cattle after their kind, of every creeping thing of the earth after his kind, two of every sort shall come unto thee, to keep them alive.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And take thou unto thee of all food that is eaten, and thou shalt gather it to thee; and it shall be for food for thee, and for them.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| Thus did Noah; according to all that God commanded him, so did he.}}<ref name=":2">{{Cite web|url=https://twitter.com/joelbaden/status/1285524127156240385|title=Genesis 6:14-7:5|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> ==Chapter 7== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And the LORD said unto Noah, Come thou and all thy house into the ark; for thee have I seen righteous before me in this generation.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| Of fowls also of the air by sevens, the male and the female; to keep seed alive upon the face of all the earth.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| For yet seven days, and I will cause it to rain upon the earth forty days and forty nights; and every living substance that I have made will I destroy from off the face of the earth.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Noah did according unto all that the LORD commanded him.}}<ref name=":2" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And Noah was six hundred years old when the flood of waters was upon the earth.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And Noah went in, and his sons, and his wife, and his sons' wives with him, into the ark, because of the waters of the flood.}}<ref name=":3">{{Cite web|url=https://twitter.com/joelbaden/status/1285910229893484545|title=Genesis 7:6-16|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#880000|Of clean beasts, and of beasts that are not clean, and of fowls,}}<ref name=":3" /> {{font|color=#888800|and of every thing that creepeth upon the earth,}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| There went in two and two unto Noah into the ark, the male and the female, as God had commanded Noah.}}<ref name=":3" /> {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And it came to pass after seven days, that the waters of the flood were upon the earth.}}<ref name=":3" /> {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| In the six hundredth year of Noah's life, in the second month, the seventeenth day of the month, the same day were all the fountains of the great deep broken up, and the windows of heaven were opened.}}<ref name=":3" /> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And the rain was upon the earth forty days and forty nights.}}<ref name=":3" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| In the selfsame day entered Noah, and Shem, and Ham, and Japheth, the sons of Noah, and Noah's wife, and the three wives of his sons with them, into the ark;}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| They, and every beast after his kind, and all the cattle after their kind, and every creeping thing that creepeth upon the earth after his kind, and every fowl after his kind, every bird of every sort.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And they went in unto Noah into the ark, two and two of all flesh, wherein is the breath of life.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And they that went in, went in male and female of all flesh, as God had commanded him:}}<ref name=":3" /> {{font|color=#000088| and the LORD shut him in}}.<ref name=":3" /> {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the flood was forty days upon the earth;}}<ref name=":4">{{Cite web|url=https://twitter.com/joelbaden/status/1286251159678849026|title=Genesis 7:17-8:5|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|color=#888800|and the waters increased, and bare up the ark, and it was lift up above the earth.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And the waters prevailed, and were increased greatly upon the earth; and the ark went upon the face of the waters.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And the waters prevailed exceedingly upon the earth; and all the high hills, that were under the whole heaven, were covered.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| Fifteen cubits upward did the waters prevail; and the mountains were covered.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And all flesh died that moved upon the earth, both of fowl, and of cattle, and of beast, and of every creeping thing that creepeth upon the earth, and every man:}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| All in whose nostrils was the breath of life, of all that was in the dry land, died.}}<ref name=":4" /> {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And every living substance was destroyed which was upon the face of the ground, both man, and cattle, and the creeping things, and the fowl of the heaven; and they were destroyed from the earth: and Noah only remained alive, and they that were with him in the ark.}}<ref name=":4" /> {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And the waters prevailed upon the earth an hundred and fifty days.}}<ref name=":4" /> ==Chapter 8== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And God remembered Noah, and every living thing, and all the cattle that was with him in the ark: and God made a wind to pass over the earth, and the waters asswaged;}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| The fountains also of the deep and the windows of heaven were stopped,}}<ref name=":4" /> {{font|color=#000088| and the rain from heaven was restrained;}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088|And the waters returned from off the earth continually:}}<ref name=":4" /> {{font|color=#888800|and after the end of the hundred and fifty days the waters were abated.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And the ark rested in the seventh month, on the seventeenth day of the month, upon the mountains of Ararat.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800|And the waters decreased continually until }}<ref name=":4" /> {{font|color=#880000|the tenth month: in the tenth month, on the first day of the month,}}<ref name=":5">{{Cite web|url=https://twitter.com/joelbaden/status/1286630319693737985|title=Genesis 8:6-14|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|color=#888800|were the tops of the mountains seen.}}<ref name=":4" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And it came to pass at the end of forty days,}}<ref name=":5" /> {{font|color=#888800|that Noah opened the window of the ark which he had made:}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And he sent forth a raven, which went forth to and fro, until the waters were dried up from off the earth.}}<ref name=":5" /> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| Also he sent forth a dove from him, to see if the waters were abated from off the face of the ground;}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| But the dove found no rest for the sole of her foot, and she returned unto him into the ark, for the waters were on the face of the whole earth: then he put forth his hand, and took her, and pulled her in unto him into the ark.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he stayed yet other seven days; and again he sent forth the dove out of the ark;}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And the dove came in to him in the evening; and, lo, in her mouth was an olive leaf pluckt off: so Noah knew that the waters were abated from off the earth.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he stayed yet other seven days; and sent forth the dove; which returned not again unto him any more.}}<ref name=":5" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And it came to pass in the six hundredth and first year, in the first month, the first day of the month, the waters were dried up from off the earth:}}<ref name=":5" /> {{font|color=#000088| and Noah removed the covering of the ark, and looked, and, behold, the face of the ground was dry.}}<ref name=":5" /> {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And in the second month, on the seven and twentieth day of the month, was the earth dried.}}<ref name=":5" /> {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And God spake unto Noah, saying,}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| Go forth of the ark, thou, and thy wife, and thy sons, and thy sons' wives with thee.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| Bring forth with thee every living thing that is with thee, of all flesh, both of fowl, and of cattle, and of every creeping thing that creepeth upon the earth; that they may breed abundantly in the earth, and be fruitful, and multiply upon the earth.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And Noah went forth, and his sons, and his wife, and his sons' wives with him:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| Every beast, every creeping thing, and every fowl, and whatsoever creepeth upon the earth, after their kinds, went forth out of the ark.}}<ref name=":6">{{Cite web|url=https://twitter.com/joelbaden/status/1286988765160972289|title=Genesis 8:15-22|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-06}}</ref> {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Noah builded an altar unto the LORD; and took of every clean beast, and of every clean fowl, and offered burnt offerings on the altar.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And the LORD smelled a sweet savour; and the LORD said in his heart, I will not again curse the ground any more for man's sake; for the imagination of man's heart is evil from his youth; neither will I again smite any more every thing living, as I have done.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| While the earth remaineth, seedtime and harvest, and cold and heat, and summer and winter, and day and night shall not cease.}}<ref name=":6" /> ==Chapter 9== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And God blessed Noah and his sons, and said unto them, Be fruitful, and multiply, and replenish the earth.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| And the fear of you and the dread of you shall be upon every beast of the earth, and upon every fowl of the air, upon all that moveth upon the earth, and upon all the fishes of the sea; into your hand are they delivered.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| But flesh with the life thereof, which is the blood thereof, shall ye not eat.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And surely your blood of your lives will I require; at the hand of every beast will I require it, and at the hand of man; at the hand of every man's brother will I require the life of man.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| Whoso sheddeth man's blood, by man shall his blood be shed: for in the image of God made he man.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And you, be ye fruitful, and multiply; bring forth abundantly in the earth, and multiply therein.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1287337660231700481|title=Genesis 9:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And God spake unto Noah, and to his sons with him, saying,}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And I, behold, I establish my covenant with you, and with your seed after you;}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And with every living creature that is with you, of the fowl, of the cattle, and of every beast of the earth with you; from all that go out of the ark, to every beast of the earth.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And God said, This is the token of the covenant which I make between me and you and every living creature that is with you, for perpetual generations:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And it shall come to pass, when I bring a cloud over the earth, that the bow shall be seen in the cloud:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And I will remember my covenant, which is between me and you and every living creature of all flesh; and the waters shall no more become a flood to destroy all flesh.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And God said unto Noah, This is the token of the covenant, which I have established between me and all flesh that is upon the earth.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1287698994727129088|title=Genesis 9:8-17|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And the sons of Noah, that went forth of the ark, were Shem, and Ham, and Japheth:}}<ref name=":7">{{Cite web|url=https://twitter.com/joelbaden/status/1288082628172820481|title=Genesis 9:18-29|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|color=#880000|and Ham is the father of Canaan.}}<ref name=":7" /> {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| These are the three sons of Noah: and of them was the whole earth overspread.}}<ref name=":7" /> {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Noah began to be an husbandman, and he planted a vineyard:}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And he drank of the wine, and was drunken; and he was uncovered within his tent.}}<ref name=":7" /> {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And}} {{font|color=#880000| Ham, the father of}}<ref name=":7" /> {{font|color=#000088|Canaan, saw the nakedness of his father, and told his two brethren without.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Shem and Japheth took a garment, and laid it upon both their shoulders, and went backward, and covered the nakedness of their father; and their faces were backward, and they saw not their father's nakedness.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And Noah awoke from his wine, and knew what his younger son had done unto him.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And he said, Cursed be Canaan; a servant of servants shall he be unto his brethren.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And he said, Blessed be the LORD God of Shem; and Canaan shall be his servant.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| God shall enlarge Japheth, and he shall dwell in the tents of Shem; and Canaan shall be his servant.}}<ref name=":7" /> {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And Noah lived after the flood three hundred and fifty years.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#888800| And all the days of Noah were nine hundred and fifty years: and he died.}}<ref name=":7" /> ==Chapter 10== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800|Now these are the generations of the sons of Noah, Shem, Ham, and Japheth: and unto them were sons born after the flood.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| The sons of Japheth; Gomer, and Magog, and Madai, and Javan, and Tubal, and Meshech, and Tiras.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And the sons of Gomer; Ashkenaz, and Riphath, and Togarmah.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And the sons of Javan; Elishah, and Tarshish, Kittim, and Dodanim.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| By these were the isles of the Gentiles divided in their lands; every one after his tongue, after their families, in their nations.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And the sons of Ham; Cush, and Mizraim, and Phut, and Canaan.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And the sons of Cush; Seba, and Havilah, and Sabtah, and Raamah, and Sabtechah: and the sons of Raamah; Sheba, and Dedan.}}<ref name=":8">{{Cite web|url=https://twitter.com/joelbaden/status/1288437150069989377|title=Genesis 10|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Cush begat Nimrod: he began to be a mighty one in the earth.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| He was a mighty hunter before the LORD: wherefore it is said, Even as Nimrod the mighty hunter before the LORD.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And the beginning of his kingdom was Babel, and Erech, and Accad, and Calneh, in the land of Shinar.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Out of that land went forth Asshur, and builded Nineveh, and the city Rehoboth, and Calah,}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And Resen between Nineveh and Calah: the same is a great city.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And Mizraim begat Ludim, and Anamim, and Lehabim, and Naphtuhim,}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Pathrusim, and Casluhim, (out of whom came Philistim,) and Caphtorim.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And Canaan begat Sidon his first born, and Heth,}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And the Jebusite, and the Amorite, and the Girgasite,}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the Hivite, and the Arkite, and the Sinite,}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And the Arvadite, and the Zemarite, and the Hamathite: and afterward were the families of the Canaanites spread abroad.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And the border of the Canaanites was from Sidon, as thou comest to Gerar, unto Gaza; as thou goest, unto Sodom, and Gomorrah, and Admah, and Zeboim, even unto Lasha.}}<ref name=":8" /> {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| These are the sons of Ham, after their families, after their tongues, in their countries, and in their nations.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Unto Shem also, the father of all the children of Eber, the brother of Japheth the elder, even to him were children born.}}<ref name=":8" /> {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| The children of Shem; Elam, and Asshur, and Arphaxad, and Lud, and Aram.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And the children of Aram; Uz, and Hul, and Gether, and Mash.}}<ref name=":8" /> {{font|size=smaller|color=#0000FF|24}}{{font|color=#880000| And Arphaxad begat Salah; and Salah begat Eber.}}<ref name=":8" /> {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And unto Eber were born two sons: the name of one was Peleg; for in his days was the earth divided; and his brother's name was Joktan.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And Joktan begat Almodad, and Sheleph, and Hazarmaveth, and Jerah,}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Hadoram, and Uzal, and Diklah,}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And Obal, and Abimael, and Sheba,}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And Ophir, and Havilah, and Jobab: all these were the sons of Joktan.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And their dwelling was from Mesha, as thou goest unto Sephar a mount of the east.}}<ref name=":8" /> {{font|size=smaller|color=#0000FF|31}}{{font|color=#888800| These are the sons of Shem, after their families, after their tongues, in their lands, after their nations.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#888800| These are the families of the sons of Noah, after their generations, in their nations: and by these were the nations divided in the earth after the flood.}}<ref name=":8" /> ==Chapter 11== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And the whole earth was of one language, and of one speech.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And it came to pass, as they journeyed from the east, that they found a plain in the land of Shinar; and they dwelt there.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And they said one to another, Go to, let us make brick, and burn them thoroughly. And they had brick for stone, and slime had they for morter.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And they said, Go to, let us build us a city and a tower, whose top may reach unto heaven; and let us make us a name, lest we be scattered abroad upon the face of the whole earth.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And the LORD came down to see the city and the tower, which the children of men builded.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And the LORD said, Behold, the people is one, and they have all one language; and this they begin to do: and now nothing will be restrained from them, which they have imagined to do.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| Go to, let us go down, and there confound their language, that they may not understand one another's speech.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| So the LORD scattered them abroad from thence upon the face of all the earth: and they left off to build the city.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| Therefore is the name of it called Babel; because the LORD did there confound the language of all the earth: and from thence did the LORD scatter them abroad upon the face of all the earth.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1288804193734729728|title=Genesis 11:1-9|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| These are the generations of Shem: Shem was an hundred years old, and begat Arphaxad two years after the flood:}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And Shem lived after he begat Arphaxad five hundred years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And Arphaxad lived five and thirty years, and begat Salah:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And Arphaxad lived after he begat Salah four hundred and three years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And Salah lived thirty years, and begat Eber:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And Salah lived after he begat Eber four hundred and three years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And Eber lived four and thirty years, and begat Peleg:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And Eber lived after he begat Peleg four hundred and thirty years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And Peleg lived thirty years, and begat Reu:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And Peleg lived after he begat Reu two hundred and nine years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And Reu lived two and thirty years, and begat Serug:}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And Reu lived after he begat Serug two hundred and seven years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| And Serug lived thirty years, and begat Nahor:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And Serug lived after he begat Nahor two hundred years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And Nahor lived nine and twenty years, and begat Terah:}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| And Nahor lived after he begat Terah an hundred and nineteen years, and begat sons and daughters.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| And Terah lived seventy years, and begat Abram,}}<ref name=":9">{{Cite web|url=https://twitter.com/joelbaden/status/1289171239412498432|title=Genesis 11:10-32|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|color=#880000| Nahor,}}<ref name=":9" /> {{font|color=#888800|and Haran.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| Now these are the generations of Terah:}}{{font|color=#888800| Terah begat Abram, }}<ref name=":9" /> {{font|color=#880000| Nahor,}}<ref name=":9" /> {{font|color=#888800|and Haran; and Haran begat Lot. }} {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And Haran died before his father Terah in the land of his nativity, in Ur of the Chaldees.}}<ref name=":9" /> {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And Abram and Nahor took them wives: the name of Abram's wife was Sarai; and the name of Nahor's wife, Milcah, the daughter of Haran, the father of Milcah, and the father of Iscah.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| But Sarai was barren; she had no child.}}<ref name=":9" /> {{font|size=smaller|color=#0000FF|31}}{{font|color=#888800| And Terah took Abram his son, and Lot the son of Haran his son's son, and Sarai his daughter in law, his son Abram's wife; and they went forth with them from Ur of the Chaldees, to go into the land of Canaan; and they came unto Haran, and dwelt there.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#888800| And the days of Terah were two hundred and five years: and Terah died in Haran.}}<ref name=":9" /> ==Chapter 12== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Now the LORD had said unto Abram, Get thee out of thy country, and from thy kindred, and from thy father's house, unto a land that I will shew thee:}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And I will make of thee a great nation, and I will bless thee, and make thy name great; and thou shalt be a blessing:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And I will bless them that bless thee, and curse him that curseth thee: and in thee shall all families of the earth be blessed.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1289891636911075328|title=Genesis 12:1-3|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| So Abram departed, as the LORD had spoken unto him; and Lot went with him:}}<ref name=":10">{{Cite web|url=https://twitter.com/joelbaden/status/1290244365164584965|title=Genesis 12:4-9|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|color=#888800| and Abram was seventy and five years old when he departed out of Haran}}. {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And Abram took Sarai his wife, and Lot his brother's son, and all their substance that they had gathered, and the souls that they had gotten in Haran; and they went forth to go into the land of Canaan; and into the land of Canaan they came.}}<ref name=":10" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Abram passed through the land unto the place of Sichem, unto the plain of Moreh. And the Canaanite was then in the land.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the LORD appeared unto Abram, and said, Unto thy seed will I give this land: and there builded he an altar unto the LORD, who appeared unto him.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And he removed from thence unto a mountain on the east of Bethel, and pitched his tent, having Bethel on the west, and Hai on the east: and there he builded an altar unto the LORD, and called upon the name of the LORD.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Abram journeyed, going on still toward the south.}}<ref name=":10" /> {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And there was a famine in the land: and Abram went down into Egypt to sojourn there; for the famine was grievous in the land.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And it came to pass, when he was come near to enter into Egypt, that he said unto Sarai his wife, Behold now, I know that thou art a fair woman to look upon:}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| Therefore it shall come to pass, when the Egyptians shall see thee, that they shall say, This is his wife: and they will kill me, but they will save thee alive.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| Say, I pray thee, thou art my sister: that it may be well with me for thy sake; and my soul shall live because of thee.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And it came to pass, that, when Abram was come into Egypt, the Egyptians beheld the woman that she was very fair.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| The princes also of Pharaoh saw her, and commended her before Pharaoh: and the woman was taken into Pharaoh's house.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And he entreated Abram well for her sake: and he had sheep, and oxen, and he asses, and menservants, and maidservants, and she asses, and camels.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the LORD plagued Pharaoh and his house with great plagues because of Sarai Abram's wife.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And Pharaoh called Abram, and said, What is this that thou hast done unto me? why didst thou not tell me that she was thy wife?}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| Why saidst thou, She is my sister? so I might have taken her to me to wife: now therefore behold thy wife, take her, and go thy way.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Pharaoh commanded his men concerning him: and they sent him away, and his wife, and all that he had.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1290613846265257985|title=Genesis 12:10-20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> ==Chapter 13== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Abram went up out of Egypt, he, and his wife, and all that he had, and Lot with him, into the south.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Abram was very rich in cattle, in silver, and in gold.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And he went on his journeys from the south even to Bethel, unto the place where his tent had been at the beginning, between Bethel and Hai;}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| Unto the place of the altar, which he had made there at the first: and there Abram called on the name of the LORD.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Lot also, which went with Abram, had flocks, and herds, and tents.}}<ref name=":11">{{Cite web|url=https://twitter.com/joelbaden/status/1290985477101821954|title=Genesis 13|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And the land was not able to bear them, that they might dwell together: for their substance was great, so that they could not dwell together.}}<ref name=":11" /> {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And there was a strife between the herdmen of Abram's cattle and the herdmen of Lot's cattle: and the Canaanite and the Perizzite dwelled then in the land.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Abram said unto Lot, Let there be no strife, I pray thee, between me and thee, and between my herdmen and thy herdmen; for we be brethren.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| Is not the whole land before thee? separate thyself, I pray thee, from me: if thou wilt take the left hand, then I will go to the right; or if thou depart to the right hand, then I will go to the left.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And Lot lifted up his eyes, and beheld all the plain of Jordan, that it was well watered every where, before the LORD destroyed Sodom and Gomorrah, even as the garden of the LORD, like the land of Egypt, as thou comest unto Zoar.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Then Lot chose him all the plain of Jordan; and Lot journeyed east:}}<ref name=":11" /> {{font|color=#888800| and they separated themselves the one from the other.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| Abram dwelled in the land of Canaan, and Lot dwelled in the cities of the plain,}}<ref name=":11" /> {{font|color=#000088|and pitched his tent toward Sodom}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| But the men of Sodom were wicked and sinners before the LORD exceedingly.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And the LORD said unto Abram, after that Lot was separated from him, Lift up now thine eyes, and look from the place where thou art northward, and southward, and eastward, and westward:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| For all the land which thou seest, to thee will I give it, and to thy seed for ever.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And I will make thy seed as the dust of the earth: so that if a man can number the dust of the earth, then shall thy seed also be numbered.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| Arise, walk through the land in the length of it and in the breadth of it; for I will give it unto thee.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| Then Abram removed his tent, and came and dwelt in the plain of Mamre, which is in Hebron, and built there an altar unto the LORD.}}<ref name=":11" /> ==Chapter 14== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And it came to pass in the days of Amraphel king of Shinar, Arioch king of Ellasar, Chedorlaomer king of Elam, and Tidal king of nations;}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| That these made war with Bera king of Sodom, and with Birsha king of Gomorrah, Shinab king of Admah, and Shemeber king of Zeboiim, and the king of Bela, which is Zoar.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| All these were joined together in the vale of Siddim, which is the salt sea.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| Twelve years they served Chedorlaomer, and in the thirteenth year they rebelled.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And in the fourteenth year came Chedorlaomer, and the kings that were with him, and smote the Rephaims in Ashteroth Karnaim, and the Zuzims in Ham, and the Emims in Shaveh Kiriathaim,}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And the Horites in their mount Seir, unto Elparan, which is by the wilderness.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And they returned, and came to Enmishpat, which is Kadesh, and smote all the country of the Amalekites, and also the Amorites, that dwelt in Hazezontamar.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And there went out the king of Sodom, and the king of Gomorrah, and the king of Admah, and the king of Zeboiim, and the king of Bela (the same is Zoar;) and they joined battle with them in the vale of Siddim;}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| With Chedorlaomer the king of Elam, and with Tidal king of nations, and Amraphel king of Shinar, and Arioch king of Ellasar; four kings with five.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And the vale of Siddim was full of slimepits; and the kings of Sodom and Gomorrah fled, and fell there; and they that remained fled to the mountain.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And they took all the goods of Sodom and Gomorrah, and all their victuals, and went their way.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1291324122304253958|title=Genesis 14:1-11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And they took Lot, Abram's brother's son, who dwelt in Sodom, and his goods, and departed.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And there came one that had escaped, and told Abram the Hebrew; for he dwelt in the plain of Mamre the Amorite, brother of Eshcol, and brother of Aner: and these were confederate with Abram.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And when Abram heard that his brother was taken captive, he armed his trained servants, born in his own house, three hundred and eighteen, and pursued them unto Dan.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And he divided himself against them, he and his servants, by night, and smote them, and pursued them unto Hobah, which is on the left hand of Damascus.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And he brought back all the goods, and also brought again his brother Lot, and his goods, and the women also, and the people.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the king of Sodom went out to meet him after his return from the slaughter of Chedorlaomer, and of the kings that were with him, at the valley of Shaveh, which is the king's dale.}}<ref name=":12">{{Cite web|url=https://twitter.com/joelbaden/status/1291699347743289345|title=Genesis 14:24|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-07}}</ref> {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Melchizedek king of Salem brought forth bread and wine: and he was the priest of the most high God.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And he blessed him, and said, Blessed be Abram of the most high God, possessor of heaven and earth:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And blessed be the most high God, which hath delivered thine enemies into thy hand. And he gave him tithes of all.}}<ref name=":12" /> {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And the king of Sodom said unto Abram, Give me the persons, and take the goods to thyself.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Abram said to the king of Sodom, I have lift up mine hand unto the LORD,}}<ref name=":12" /> {{font|color=#880000|the most high God, the possessor of heaven and earth,}}<ref name=":12" /> {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| That I will not take from a thread even to a shoelatchet, and that I will not take any thing that is thine, lest thou shouldest say, I have made Abram rich:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| Save only that which the young men have eaten, and the portion of the men which went with me, Aner, Eshcol, and Mamre; let them take their portion.}}<ref name=":12" /> ==Chapter 15== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| After these things the word of the LORD came unto Abram in a vision, saying, Fear not, Abram: I am thy shield, and thy exceeding great reward.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And Abram said,}}<ref name=":14" /> {{font|color=#880000|LORD God,}}<ref name=":14" /> {{font|color=#008888|what wilt thou give me, seeing I go childless, and the steward of my house is this Eliezer of Damascus?}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And Abram said, Behold, to me thou hast given no seed: and, lo, one born in my house is mine heir.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| And, behold, the word of the LORD came unto him, saying, This shall not be thine heir; but he that shall come forth out of thine own bowels shall be thine heir.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And he brought him forth abroad, and said, Look now toward heaven, and tell the stars, if thou be able to number them: and he said unto him, So shall thy seed be.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And he believed in the LORD; and he counted it to him for righteousness.}}<ref name=":14">{{Cite web|url=https://twitter.com/joelbaden/status/1292075583564853248|title=Genesis 15:1-6|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-08}}</ref> {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| And he said unto him, I am the}} {{font|color=#880000|LORD that brought thee out of Ur of the Chaldees,}}<ref>{{Cite book|url=https://www.worldcat.org/oclc/809789652|title=The promise to the Patriarchs|last=Baden|first=Joel S.|date=2013|publisher=Oxford University Press|isbn=978-0-19-989824-4|location=Oxford|oclc=809789652}}</ref> {{font|color=#008888|to give thee this land to inherit it.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And he said,}}<ref name=":13" /> {{font|color=#880000|LORD God,}}<ref name=":14" /> {{font|color=#008888|whereby shall I know that I shall inherit it?}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And he said unto him, Take me an heifer of three years old, and a she goat of three years old, and a ram of three years old, and a turtledove, and a young pigeon.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And he took unto him all these, and divided them in the midst, and laid each piece one against another: but the birds divided he not.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And when the fowls came down upon the carcases, Abram drove them away.}}<ref name=":13">{{Cite web|url=https://twitter.com/joelbaden/status/1292435923792203781|title=Genesis 15:7-20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-09}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#880000| And when the sun was going down, a deep sleep fell upon Abram; and, lo, an horror of great darkness fell upon him.}}<ref name=":13" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And he said unto Abram, Know of a surety that thy seed shall be a stranger in a land that is not theirs,}}<ref name=":13" /> {{font|color=#880000|and shall serve them; and they shall afflict them four hundred years;}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#880000| And also that nation, whom they shall serve, will I judge: and afterward shall they come out with great substance.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#880000| And thou shalt go to thy fathers in peace; thou shalt be buried in a good old age.}}<ref name=":13" /> {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| But in the fourth generation they shall come hither again: for the iniquity of the Amorites is not yet full.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And it came to pass, that, when the sun went down, and it was dark, behold a smoking furnace, and a burning lamp that passed between those pieces.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| In the same day the LORD made a covenant with Abram, saying, Unto thy seed have I given this land, from the river of Egypt unto the great river, the river Euphrates:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| The Kenites, and the Kenizzites, and the Kadmonites,}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And the Hittites, and the Perizzites, and the Rephaims,}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And the Amorites, and the Canaanites, and the Girgashites, and the Jebusites.}}<ref name=":13" /> ==Chapter 16== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| Now Sarai Abram's wife bare him no children: and she had an handmaid, an Egyptian, whose name was Hagar.}}<ref name=":15">{{Cite web|url=https://twitter.com/joelbaden/status/1292791743717482497|title=Genesis 16:1-6|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Sarai said unto Abram, Behold now, the LORD hath restrained me from bearing: I pray thee, go in unto my maid; it may be that I may obtain children by her. And Abram hearkened to the voice of Sarai.}}<ref name=":15" /> {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Sarai Abram's wife took Hagar her maid the Egyptian, after Abram had dwelt ten years in the land of Canaan, and gave her to her husband Abram to be his wife.}}<ref name=":15" /> {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And he went in unto Hagar, and she conceived: and when she saw that she had conceived, her mistress was despised in her eyes.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Sarai said unto Abram, My wrong be upon thee: I have given my maid into thy bosom; and when she saw that she had conceived, I was despised in her eyes: the LORD judge between me and thee.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| But Abram said unto Sarai, Behold, thy maid is in thy hand; do to her as it pleaseth thee. And when Sarai dealt hardly with her, she fled from her face.}}<ref name=":15" /> {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the angel of the LORD found her by a fountain of water in the wilderness, by the fountain in the way to Shur.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And he said, Hagar, Sarai's maid, whence camest thou? and whither wilt thou go? And she said, I flee from the face of my mistress Sarai.}}<ref name=":16">{{Cite web|url=https://twitter.com/joelbaden/status/1293150235368263680|title=Genesis 16:7-16|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> {{font|size=smaller|color=#0000FF|9}}{{font|color=#880000| And the angel of the LORD said unto her, Return to thy mistress, and submit thyself under her hands.}}<ref name=":16" /> {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And the angel of the LORD said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And the angel of the LORD said unto her, Behold, thou art with child, and shalt bear a son, and shalt call his name Ishmael; because the LORD hath heard thy affliction.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he will be a wild man; his hand will be against every man, and every man's hand against him; and he shall dwell in the presence of all his brethren.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And she called the name of the LORD that spake unto her, Thou God seest me: for she said, Have I also here looked after him that seeth me?}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Wherefore the well was called Beerlahairoi; behold, it is between Kadesh and Bered.}}<ref name=":16" /> {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And Hagar bare Abram a son: and Abram called his son's name, which Hagar bare, Ishmael.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And Abram was fourscore and six years old, when Hagar bare Ishmael to Abram.}}<ref name=":16" /> ==Chapter 17== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And when Abram was ninety years old and nine, the LORD appeared to Abram, and said unto him, I am the Almighty God; walk before me, and be thou perfect.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| And I will make my covenant between me and thee, and will multiply thee exceedingly.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Abram fell on his face: and God talked with him, saying,}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| As for me, behold, my covenant is with thee, and thou shalt be a father of many nations.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| Neither shall thy name any more be called Abram, but thy name shall be Abraham; for a father of many nations have I made thee.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And I will make thee exceeding fruitful, and I will make nations of thee, and kings shall come out of thee.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And I will establish my covenant between me and thee and thy seed after thee in their generations for an everlasting covenant, to be a God unto thee, and to thy seed after thee.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And I will give unto thee, and to thy seed after thee, the land wherein thou art a stranger, all the land of Canaan, for an everlasting possession; and I will be their God.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1293526590899576833|title=Genesis 17:1-8|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And God said unto Abraham, Thou shalt keep my covenant therefore, thou, and thy seed after thee in their generations.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| This is my covenant, which ye shall keep, between me and you and thy seed after thee; Every man child among you shall be circumcised.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And ye shall circumcise the flesh of your foreskin; and it shall be a token of the covenant betwixt me and you.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And he that is eight days old shall be circumcised among you, every man child in your generations, he that is born in the house, or bought with money of any stranger, which is not of thy seed.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| He that is born in thy house, and he that is bought with thy money, must needs be circumcised: and my covenant shall be in your flesh for an everlasting covenant.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And the uncircumcised man child whose flesh of his foreskin is not circumcised, that soul shall be cut off from his people; he hath broken my covenant.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1293884546505736192|title=Genesis 17:9-14|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And God said unto Abraham, As for Sarai thy wife, thou shalt not call her name Sarai, but Sarah shall her name be.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And I will bless her, and give thee a son also of her: yea, I will bless her, and she shall be a mother of nations; kings of people shall be of her.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| Then Abraham fell upon his face, and laughed, and said in his heart, Shall a child be born unto him that is an hundred years old? and shall Sarah, that is ninety years old, bear?}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And Abraham said unto God, O that Ishmael might live before thee!}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And God said, Sarah thy wife shall bear thee a son indeed; and thou shalt call his name Isaac: and I will establish my covenant with him for an everlasting covenant, and with his seed after him.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And as for Ishmael, I have heard thee: Behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly; twelve princes shall he beget, and I will make him a great nation.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| But my covenant will I establish with Isaac, which Sarah shall bear unto thee at this set time in the next year.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| And he left off talking with him, and God went up from Abraham.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And Abraham took Ishmael his son, and all that were born in his house, and all that were bought with his money, every male among the men of Abraham's house; and circumcised the flesh of their foreskin in the selfsame day, as God had said unto him.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And Abraham was ninety years old and nine, when he was circumcised in the flesh of his foreskin.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| And Ishmael his son was thirteen years old, when he was circumcised in the flesh of his foreskin.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| In the selfsame day was Abraham circumcised, and Ishmael his son.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| And all the men of his house, born in the house, and bought with money of the stranger, were circumcised with him.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1294238305920471041|title=Genesis 17:15-27|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> ==Chapter 18== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And the LORD appeared unto him in the plains of Mamre: and he sat in the tent door in the heat of the day;}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he lift up his eyes and looked, and, lo, three men stood by him: and when he saw them, he ran to meet them from the tent door, and bowed himself toward the ground,}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And said, My LORD, if now I have found favour in thy sight, pass not away, I pray thee, from thy servant:}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| Let a little water, I pray you, be fetched, and wash your feet, and rest yourselves under the tree:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And I will fetch a morsel of bread, and comfort ye your hearts; after that ye shall pass on: for therefore are ye come to your servant. And they said, So do, as thou hast said.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Abraham hastened into the tent unto Sarah, and said, Make ready quickly three measures of fine meal, knead it, and make cakes upon the hearth.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Abraham ran unto the herd, and fetcht a calf tender and good, and gave it unto a young man; and he hasted to dress it.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And he took butter, and milk, and the calf which he had dressed, and set it before them; and he stood by them under the tree, and they did eat.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1294608588820840448|title=Genesis 18:1-8|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-10}}</ref> {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And they said unto him, Where is Sarah thy wife? And he said, Behold, in the tent.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he said, I will certainly return unto thee according to the time of life; and, lo, Sarah thy wife shall have a son. And Sarah heard it in the tent door, which was behind him.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Now Abraham and Sarah were old and well stricken in age; and it ceased to be with Sarah after the manner of women.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| Therefore Sarah laughed within herself, saying, After I am waxed old shall I have pleasure, my lord being old also?}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And the LORD said unto Abraham, Wherefore did Sarah laugh, saying, Shall I of a surety bear a child, which am old?}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Is any thing too hard for the LORD? At the time appointed I will return unto thee, according to the time of life, and Sarah shall have a son.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| Then Sarah denied, saying, I laughed not; for she was afraid. And he said, Nay; but thou didst laugh.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1294967646119763968|title=Genesis 18:9-15|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And the men rose up from thence, and looked toward Sodom: and Abraham went with them to bring them on the way.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the LORD said, Shall I hide from Abraham that thing which I do;}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| Seeing that Abraham shall surely become a great and mighty nation, and all the nations of the earth shall be blessed in him?}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| For I know him, that he will command his children and his household after him, and they shall keep the way of the LORD, to do justice and judgment; that the LORD may bring upon Abraham that which he hath spoken of him.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And the LORD said, Because the cry of Sodom and Gomorrah is great, and because their sin is very grievous;}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| I will go down now, and see whether they have done altogether according to the cry of it, which is come unto me; and if not, I will know.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And the men turned their faces from thence, and went toward Sodom: but Abraham stood yet before the LORD.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Abraham drew near, and said, Wilt thou also destroy the righteous with the wicked?}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| Peradventure there be fifty righteous within the city: wilt thou also destroy and not spare the place for the fifty righteous that are therein?}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| That be far from thee to do after this manner, to slay the righteous with the wicked: and that the righteous should be as the wicked, that be far from thee: Shall not the Judge of all the earth do right?}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And the LORD said, If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Abraham answered and said, Behold now, I have taken upon me to speak unto the LORD, which am but dust and ashes:}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| Peradventure there shall lack five of the fifty righteous: wilt thou destroy all the city for lack of five? And he said, If I find there forty and five, I will not destroy it.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And he spake unto him yet again, and said, Peradventure there shall be forty found there. And he said, I will not do it for forty's sake.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And he said unto him, Oh let not the LORD be angry, and I will speak: Peradventure there shall thirty be found there. And he said, I will not do it, if I find thirty there.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he said, Behold now, I have taken upon me to speak unto the LORD: Peradventure there shall be twenty found there. And he said, I will not destroy it for twenty's sake.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And he said, Oh let not the LORD be angry, and I will speak yet but this once: Peradventure ten shall be found there. And he said, I will not destroy it for ten's sake.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And the LORD went his way, as soon as he had left communing with Abraham: and Abraham returned unto his place.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1295327194139459584|title=Genesis 18:16-33|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> ==Chapter 19== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And there came two angels to Sodom at even; and Lot sat in the gate of Sodom: and Lot seeing them rose up to meet them; and he bowed himself with his face toward the ground;}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he said, Behold now, my lords, turn in, I pray you, into your servant's house, and tarry all night, and wash your feet, and ye shall rise up early, and go on your ways. And they said, Nay; but we will abide in the street all night.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And he pressed upon them greatly; and they turned in unto him, and entered into his house; and he made them a feast, and did bake unleavened bread, and they did eat.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| But before they lay down, the men of the city, even the men of Sodom, compassed the house round, both old and young, all the people from every quarter:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And they called unto Lot, and said unto him, Where are the men which came in to thee this night? bring them out unto us, that we may know them.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Lot went out at the door unto them, and shut the door after him,}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And said, I pray you, brethren, do not so wickedly.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| Behold now, I have two daughters which have not known man; let me, I pray you, bring them out unto you, and do ye to them as is good in your eyes: only unto these men do nothing; for therefore came they under the shadow of my roof.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And they said, Stand back. And they said again, This one fellow came in to sojourn, and he will needs be a judge: now will we deal worse with thee, than with them. And they pressed sore upon the man, even Lot, and came near to break the door.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| But the men put forth their hand, and pulled Lot into the house to them, and shut to the door.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And they smote the men that were at the door of the house with blindness, both small and great: so that they wearied themselves to find the door.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1295681772240490498|title=Genesis 19-1:11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And the men said unto Lot, Hast thou here any besides? son in law, and thy sons, and thy daughters, and whatsoever thou hast in the city, bring them out of this place:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| For we will destroy this place, because the cry of them is waxen great before the face of the LORD; and the LORD hath sent us to destroy it.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Lot went out, and spake unto his sons in law, which married his daughters, and said, Up, get you out of this place; for the LORD will destroy this city. But he seemed as one that mocked unto his sons in law.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And when the morning arose, then the angels hastened Lot, saying, Arise, take thy wife, and thy two daughters, which are here; lest thou be consumed in the iniquity of the city.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And while he lingered, the men laid hold upon his hand, and upon the hand of his wife, and upon the hand of his two daughters; the LORD being merciful unto him: and they brought him forth, and set him without the city.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And it came to pass, when they had brought them forth abroad, that he said, Escape for thy life; look not behind thee, neither stay thou in all the plain; escape to the mountain, lest thou be consumed.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And Lot said unto them, Oh, not so, my LORD:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| Behold now, thy servant hath found grace in thy sight, and thou hast magnified thy mercy, which thou hast shewed unto me in saving my life; and I cannot escape to the mountain, lest some evil take me, and I die:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| Behold now, this city is near to flee unto, and it is a little one: Oh, let me escape thither, (is it not a little one?) and my soul shall live.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And he said unto him, See, I have accepted thee concerning this thing also, that I will not overthrow this city, for the which thou hast spoken.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| Haste thee, escape thither; for I cannot do anything till thou be come thither. Therefore the name of the city was called Zoar.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| The sun was risen upon the earth when Lot entered into Zoar.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| Then the LORD rained upon Sodom and upon Gomorrah brimstone and fire from the LORD out of heaven;}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And he overthrew those cities, and all the plain, and all the inhabitants of the cities, and that which grew upon the ground.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| But his wife looked back from behind him, and she became a pillar of salt.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Abraham gat up early in the morning to the place where he stood before the LORD:}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And he looked toward Sodom and Gomorrah, and toward all the land of the plain, and beheld, and, lo, the smoke of the country went up as the smoke of a furnace.}}<ref name=":17">{{Cite web|url=https://twitter.com/joelbaden/status/1296045431487696896|title=Genesis 19:12-29|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|29}}{{font|color=#888800| And it came to pass, when God destroyed the cities of the plain, that God remembered Abraham, and sent Lot out of the midst of the overthrow, when he overthrew the cities in the which Lot dwelt.}}<ref name=":17" /> {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And Lot went up out of Zoar, and dwelt in the mountain, and his two daughters with him; for he feared to dwell in Zoar: and he dwelt in a cave, he and his two daughters.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And the firstborn said unto the younger, Our father is old, and there is not a man in the earth to come in unto us after the manner of all the earth:}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| Come, let us make our father drink wine, and we will lie with him, that we may preserve seed of our father.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And they made their father drink wine that night: and the firstborn went in, and lay with her father; and he perceived not when she lay down, nor when she arose.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And it came to pass on the morrow, that the firstborn said unto the younger, Behold, I lay yesternight with my father: let us make him drink wine this night also; and go thou in, and lie with him, that we may preserve seed of our father.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And they made their father drink wine that night also: and the younger arose, and lay with him; and he perceived not when she lay down, nor when she arose.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#000088| Thus were both the daughters of Lot with child by their father.}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#000088| And the first born bare a son, and called his name Moab: the same is the father of the Moabites unto this day.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#000088| And the younger, she also bare a son, and called his name Benammi: the same is the father of the children of Ammon unto this day.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1296422821967650817|title=Genesis 19:30-38|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> ==Chapter 20== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And Abraham journeyed from thence toward the south country, and dwelled between Kadesh and Shur, and sojourned in Gerar.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And Abraham said of Sarah his wife, She is my sister: and Abimelech king of Gerar sent, and took Sarah.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| But God came to Abimelech in a dream by night, and said to him, Behold, thou art but a dead man, for the woman which thou hast taken; for she is a man's wife.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| But Abimelech had not come near her: and he said, LORD, wilt thou slay also a righteous nation?}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| Said he not unto me, She is my sister? and she, even she herself said, He is my brother: in the integrity of my heart and innocency of my hands have I done this.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And God said unto him in a dream, Yea, I know that thou didst this in the integrity of thy heart; for I also withheld thee from sinning against me: therefore suffered I thee not to touch her.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| Now therefore restore the man his wife; for he is a prophet, and he shall pray for thee, and thou shalt live: and if thou restore her not, know thou that thou shalt surely die, thou, and all that are thine.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| Therefore Abimelech rose early in the morning, and called all his servants, and told all these things in their ears: and the men were sore afraid.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| Then Abimelech called Abraham, and said unto him, What hast thou done unto us? and what have I offended thee, that thou hast brought on me and on my kingdom a great sin? thou hast done deeds unto me that ought not to be done.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And Abimelech said unto Abraham, What sawest thou, that thou hast done this thing?}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And Abraham said, Because I thought, Surely the fear of God is not in this place; and they will slay me for my wife's sake.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And yet indeed she is my sister; she is the daughter of my father, but not the daughter of my mother; and she became my wife.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And it came to pass, when God caused me to wander from my father's house, that I said unto her, This is thy kindness which thou shalt shew unto me; at every place whither we shall come, say of me, He is my brother.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And Abimelech took sheep, and oxen, and menservants, and womenservants, and gave them unto Abraham, and restored him Sarah his wife.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| And Abimelech said, Behold, my land is before thee: dwell where it pleaseth thee.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| And unto Sarah he said, Behold, I have given thy brother a thousand pieces of silver: behold, he is to thee a covering of the eyes, unto all that are with thee, and with all other: thus she was reproved.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| So Abraham prayed unto God: and God healed Abimelech, and his wife, and his maidservants; and they bare children.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| For the LORD had fast closed up all the wombs of the house of Abimelech, because of Sarah Abraham's wife.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1296769585891414018|title=Genesis 20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> ==Chapter 21== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And the LORD visited Sarah as he had said, and the LORD did unto Sarah as he had spoken.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| For Sarah conceived, and bare Abraham a son in his old age,}}<ref name=":18">{{Cite web|url=https://twitter.com/joelbaden/status/1297141601723199488|title=Genesis 21:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|color=#888800| at the set time of which God had spoken to him}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Abraham called the name of his son that was born unto him, whom Sarah bare to him, Isaac.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And Abraham circumcised his son Isaac being eight days old, as God had commanded him.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And Abraham was an hundred years old, when his son Isaac was born unto him.}}<ref name=":18" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And Sarah said, God hath made me to laugh, so that all that hear will laugh with me.}}<ref name=":18" /> {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And she said, Who would have said unto Abraham, that Sarah should have given children suck? for I have born him a son in his old age.}}<ref name=":18" /> {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And the child grew, and was weaned: and Abraham made a great feast the same day that Isaac was weaned.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And Sarah saw the son of Hagar the Egyptian, which she had born unto Abraham, mocking.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| Wherefore she said unto Abraham, Cast out this bondwoman and her son: for the son of this bondwoman shall not be heir with my son, even with Isaac.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And the thing was very grievous in Abraham's sight because of his son.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And God said unto Abraham, Let it not be grievous in thy sight because of the lad, and because of thy bondwoman; in all that Sarah hath said unto thee, hearken unto her voice; for in Isaac shall thy seed be called.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And also of the son of the bondwoman will I make a nation, because he is thy seed.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And Abraham rose up early in the morning, and took bread, and a bottle of water, and gave it unto Hagar, putting it on her shoulder, and the child, and sent her away: and she departed, and wandered in the wilderness of Beersheba.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| And the water was spent in the bottle, and she cast the child under one of the shrubs.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| And she went, and sat her down over against him a good way off, as it were a bow shot: for she said, Let me not see the death of the child. And she sat over against him, and lift up her voice, and wept.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And God heard the voice of the lad; and the angel of God called to Hagar out of heaven, and said unto her, What aileth thee, Hagar? fear not; for God hath heard the voice of the lad where he is.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| Arise, lift up the lad, and hold him in thine hand; for I will make him a great nation.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And God opened her eyes, and she saw a well of water; and she went, and filled the bottle with water, and gave the lad drink.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And God was with the lad; and he grew, and dwelt in the wilderness, and became an archer.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And he dwelt in the wilderness of Paran: and his mother took him a wife out of the land of Egypt.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1297514467715162123|title=Genesis 21:8-21|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And it came to pass at that time, that Abimelech and Phichol the chief captain of his host spake unto Abraham, saying, God is with thee in all that thou doest:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| Now therefore swear unto me here by God that thou wilt not deal falsely with me, nor with my son, nor with my son's son: but according to the kindness that I have done unto thee, thou shalt do unto me, and to the land wherein thou hast sojourned.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And Abraham said, I will swear.}}<ref name=":19">{{Cite web|url=https://twitter.com/joelbaden/status/1297867597904580610|title=Genesis 21:22-34|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| And Abraham reproved Abimelech because of a well of water, which Abimelech's servants had violently taken away.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#008888| And Abimelech said, I wot not who hath done this thing: neither didst thou tell me, neither yet heard I of it, but to day.}}<ref name=":19" /> {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Abraham took sheep and oxen, and gave them unto Abimelech; and both of them made a covenant.}}<ref name=":19" /> {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| And Abraham set seven ewe lambs of the flock by themselves.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#008888| And Abimelech said unto Abraham, What mean these seven ewe lambs which thou hast set by themselves?}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#008888| And he said, For these seven ewe lambs shalt thou take of my hand, that they may be a witness unto me, that I have digged this well.}}<ref name=":19" /> {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| Wherefore he called that place Beersheba; because there they sware both of them.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| Thus they made a covenant at Beersheba: then Abimelech rose up, and Phichol the chief captain of his host, and they returned into the land of the Philistines.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And Abraham planted a grove in Beersheba, and called there on the name of the LORD, the everlasting God.}}<ref name=":19" /> {{font|size=smaller|color=#0000FF|34}}{{font|color=#008888| And Abraham sojourned in the Philistines' land many days.}}<ref name=":19" /> ==Chapter 22== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And it came to pass after these things, that God did tempt Abraham, and said unto him, Abraham: and he said, Behold, here I am.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And he said, Take now thy son, thine only son Isaac, whom thou lovest, and get thee into the land of Moriah; and offer him there for a burnt offering upon one of the mountains which I will tell thee of.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And Abraham rose up early in the morning, and saddled his ass, and took two of his young men with him, and Isaac his son, and clave the wood for the burnt offering, and rose up, and went unto the place of which God had told him.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| Then on the third day Abraham lifted up his eyes, and saw the place afar off.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And Abraham said unto his young men, Abide ye here with the ass; and I and the lad will go yonder and worship, and I shall come again to you.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And Abraham took the wood of the burnt offering, and laid it upon Isaac his son; and he took the fire in his hand, and a knife; and they went both of them together.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| And Isaac spake unto Abraham his father, and said, My father: and he said, Here am I, my son. And he said, Behold the fire and the wood: but where is the lamb for a burnt offering?}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And Abraham said, My son, God will provide himself a lamb for a burnt offering: so they went both of them together.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And they came to the place which God had told him of; and Abraham built an altar there, and laid the wood in order, and bound Isaac his son, and laid him on the altar upon the wood.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And Abraham stretched forth his hand, and took the knife to slay his son.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And the angel of the LORD called unto him out of heaven, and said, Abraham, Abraham: and he said, Here am I.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And he said, Lay not thine hand upon the lad, neither do thou any thing unto him: for now I know that thou fearest God, seeing thou hast not withheld thy son, thine only son from me.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And Abraham lifted up his eyes, and looked, and behold behind him a ram caught in a thicket by his horns: and Abraham went and took the ram, and offered him up for a burnt offering in the stead of his son.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And Abraham called the name of that place Jehovahjireh: as it is said to this day, In the mount of the LORD it shall be seen.}}<ref name=":20">{{Cite web|url=https://twitter.com/joelbaden/status/1298235350809280518|title=Genesis 22:1-19|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|15}}{{font|color=#880000| And the angel of the LORD called unto Abraham out of heaven the second time,}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#880000| And said, By myself have I sworn, saith the LORD, for because thou hast done this thing, and hast not withheld thy son, thine only son:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#880000| That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore; and thy seed shall possess the gate of his enemies;}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#880000| And in thy seed shall all the nations of the earth be blessed; because thou hast obeyed my voice.}}<ref name=":20" /> {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| So Abraham returned unto his young men, and they rose up and went together to Beersheba; and Abraham dwelt at Beersheba.}}<ref name=":20" /> {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And it came to pass after these things, that it was told Abraham, saying, Behold, Milcah, she hath also born children unto thy brother Nahor;}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Huz his firstborn, and Buz his brother, and Kemuel the father of Aram,}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Chesed, and Hazo, and Pildash, and Jidlaph, and Bethuel.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Bethuel begat Rebekah: these eight Milcah did bear to Nahor, Abraham's brother.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And his concubine, whose name was Reumah, she bare also Tebah, and Gaham, and Thahash, and Maachah.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1298595290937724928|title=Genesis 22:20-24|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> ==Chapter 23== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And Sarah was an hundred and seven and twenty years old: these were the years of the life of Sarah.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| And Sarah died in Kirjatharba; the same is Hebron in the land of Canaan: and Abraham came to mourn for Sarah, and to weep for her.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Abraham stood up from before his dead, and spake unto the sons of Heth, saying,}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| I am a stranger and a sojourner with you: give me a possession of a buryingplace with you, that I may bury my dead out of my sight.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And the children of Heth answered Abraham, saying unto him,}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| Hear us, my lord: thou art a mighty prince among us: in the choice of our sepulchres bury thy dead; none of us shall withhold from thee his sepulchre, but that thou mayest bury thy dead.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And Abraham stood up, and bowed himself to the people of the land, even to the children of Heth.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And he communed with them, saying, If it be your mind that I should bury my dead out of my sight; hear me, and intreat for me to Ephron the son of Zohar,}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| That he may give me the cave of Machpelah, which he hath, which is in the end of his field; for as much money as it is worth he shall give it me for a possession of a buryingplace amongst you.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And Ephron dwelt among the children of Heth: and Ephron the Hittite answered Abraham in the audience of the children of Heth, even of all that went in at the gate of his city, saying,}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| Nay, my lord, hear me: the field give I thee, and the cave that is therein, I give it thee; in the presence of the sons of my people give I it thee: bury thy dead.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And Abraham bowed down himself before the people of the land.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And he spake unto Ephron in the audience of the people of the land, saying, But if thou wilt give it, I pray thee, hear me: I will give thee money for the field; take it of me, and I will bury my dead there.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And Ephron answered Abraham, saying unto him,}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| My lord, hearken unto me: the land is worth four hundred shekels of silver; what is that betwixt me and thee? bury therefore thy dead.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And Abraham hearkened unto Ephron; and Abraham weighed to Ephron the silver, which he had named in the audience of the sons of Heth, four hundred shekels of silver, current money with the merchant.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And the field of Ephron, which was in Machpelah, which was before Mamre, the field, and the cave which was therein, and all the trees that were in the field, that were in all the borders round about, were made sure}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| Unto Abraham for a possession in the presence of the children of Heth, before all that went in at the gate of his city.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And after this, Abraham buried Sarah his wife in the cave of the field of Machpelah before Mamre: the same is Hebron in the land of Canaan.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And the field, and the cave that is therein, were made sure unto Abraham for a possession of a buryingplace by the sons of Heth.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1298953696563335174|title=Genesis 23|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> ==Chapter 24== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Abraham was old, and well stricken in age: and the LORD had blessed Abraham in all things.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Abraham said unto his eldest servant of his house, that ruled over all that he had, Put, I pray thee, thy hand under my thigh:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And I will make thee swear by the LORD, the God of heaven, and the God of the earth, that thou shalt not take a wife unto my son of the daughters of the Canaanites, among whom I dwell:}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| But thou shalt go unto my country, and to my kindred, and take a wife unto my son Isaac.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And the servant said unto him, Peradventure the woman will not be willing to follow me unto this land: must I needs bring thy son again unto the land from whence thou camest?}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Abraham said unto him, Beware thou that thou bring not my son thither again.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| The LORD God of heaven, which took me from my father's house, and from the land of my kindred, and which spake unto me, and that sware unto me, saying, Unto thy seed will I give this land; he shall send his angel before thee, and thou shalt take a wife unto my son from thence.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1299298481912320002|title=Genesis 24:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And if the woman will not be willing to follow thee, then thou shalt be clear from this my oath: only bring not my son thither again.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And the servant put his hand under the thigh of Abraham his master, and sware to him concerning that matter.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And the servant took ten camels of the camels of his master, and departed; for all the goods of his master were in his hand: and he arose, and went to Mesopotamia, unto the city of Nahor.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And he made his camels to kneel down without the city by a well of water at the time of the evening, even the time that women go out to draw water.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he said, O LORD God of my master Abraham, I pray thee, send me good speed this day, and shew kindness unto my master Abraham.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| Behold, I stand here by the well of water; and the daughters of the men of the city come out to draw water:}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And let it come to pass, that the damsel to whom I shall say, Let down thy pitcher, I pray thee, that I may drink; and she shall say, Drink, and I will give thy camels drink also: let the same be she that thou hast appointed for thy servant Isaac; and thereby shall I know that thou hast shewed kindness unto my master.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And it came to pass, before he had done speaking, that, behold, Rebekah came out, who was born to Bethuel, son of Milcah, the wife of Nahor, Abraham's brother, with her pitcher upon her shoulder.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And the damsel was very fair to look upon, a virgin, neither had any man known her: and she went down to the well, and filled her pitcher, and came up.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the servant ran to meet her, and said, Let me, I pray thee, drink a little water of thy pitcher.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And she said, Drink, my lord: and she hasted, and let down her pitcher upon her hand, and gave him drink.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And when she had done giving him drink, she said, I will draw water for thy camels also, until they have done drinking.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And she hasted, and emptied her pitcher into the trough, and ran again unto the well to draw water, and drew for all his camels.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And the man wondering at her held his peace, to wit whether the LORD had made his journey prosperous or not.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And it came to pass, as the camels had done drinking, that the man took a golden earring of half a shekel weight, and two bracelets for her hands of ten shekels weight of gold;}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And said, Whose daughter art thou? tell me, I pray thee: is there room in thy father's house for us to lodge in?}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And she said unto him, I am the daughter of Bethuel the son of Milcah, which she bare unto Nahor.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| She said moreover unto him, We have both straw and provender enough, and room to lodge in.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And the man bowed down his head, and worshipped the LORD.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And he said, Blessed be the LORD God of my master Abraham, who hath not left destitute my master of his mercy and his truth: I being in the way, the LORD led me to the house of my master's brethren.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And the damsel ran, and told them of her mother's house these things.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And Rebekah had a brother, and his name was Laban: and Laban ran out unto the man, unto the well.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And it came to pass, when he saw the earring and bracelets upon his sister's hands, and when he heard the words of Rebekah his sister, saying, Thus spake the man unto me; that he came unto the man; and, behold, he stood by the camels at the well.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he said, Come in, thou blessed of the LORD; wherefore standest thou without? for I have prepared the house, and room for the camels.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And the man came into the house: and he ungirded his camels, and gave straw and provender for the camels, and water to wash his feet, and the men's feet that were with him.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And there was set meat before him to eat: but he said, I will not eat, until I have told mine errand. And he said, Speak on.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And he said, I am Abraham's servant.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And the LORD hath blessed my master greatly; and he is become great: and he hath given him flocks, and herds, and silver, and gold, and menservants, and maidservants, and camels, and asses.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#000088| And Sarah my master's wife bare a son to my master when she was old: and unto him hath he given all that he hath.}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#000088| And my master made me swear, saying, Thou shalt not take a wife to my son of the daughters of the Canaanites, in whose land I dwell:}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#000088| But thou shalt go unto my father's house, and to my kindred, and take a wife unto my son.}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#000088| And I said unto my master, Peradventure the woman will not follow me.}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#000088| And he said unto me, The LORD, before whom I walk, will send his angel with thee, and prosper thy way; and thou shalt take a wife for my son of my kindred, and of my father's house:}} {{font|size=smaller|color=#0000FF|41}}{{font|color=#000088| Then shalt thou be clear from this my oath, when thou comest to my kindred; and if they give not thee one, thou shalt be clear from my oath.}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#000088| And I came this day unto the well, and said, O LORD God of my master Abraham, if now thou do prosper my way which I go:}} {{font|size=smaller|color=#0000FF|43}}{{font|color=#000088| Behold, I stand by the well of water; and it shall come to pass, that when the virgin cometh forth to draw water, and I say to her, Give me, I pray thee, a little water of thy pitcher to drink;}} {{font|size=smaller|color=#0000FF|44}}{{font|color=#000088| And she say to me, Both drink thou, and I will also draw for thy camels: let the same be the woman whom the LORD hath appointed out for my master's son.}} {{font|size=smaller|color=#0000FF|45}}{{font|color=#000088| And before I had done speaking in mine heart, behold, Rebekah came forth with her pitcher on her shoulder; and she went down unto the well, and drew water: and I said unto her, Let me drink, I pray thee.}} {{font|size=smaller|color=#0000FF|46}}{{font|color=#000088| And she made haste, and let down her pitcher from her shoulder, and said, Drink, and I will give thy camels drink also: so I drank, and she made the camels drink also.}} {{font|size=smaller|color=#0000FF|47}}{{font|color=#000088| And I asked her, and said, Whose daughter art thou? And she said, The daughter of Bethuel, Nahor's son, whom Milcah bare unto him: and I put the earring upon her face, and the bracelets upon her hands.}} {{font|size=smaller|color=#0000FF|48}}{{font|color=#000088| And I bowed down my head, and worshipped the LORD, and blessed the LORD God of my master Abraham, which had led me in the right way to take my master's brother's daughter unto his son.}} {{font|size=smaller|color=#0000FF|49}}{{font|color=#000088| And now if ye will deal kindly and truly with my master, tell me: and if not, tell me; that I may turn to the right hand, or to the left.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1299696566756945920|title=Genesis 24:10-49|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|50}}{{font|color=#000088| Then Laban and Bethuel answered and said, The thing proceedeth from the LORD: we cannot speak unto thee bad or good.}} {{font|size=smaller|color=#0000FF|51}}{{font|color=#000088| Behold, Rebekah is before thee, take her, and go, and let her be thy master's son's wife, as the LORD hath spoken.}} {{font|size=smaller|color=#0000FF|52}}{{font|color=#000088| And it came to pass, that, when Abraham's servant heard their words, he worshipped the LORD, bowing himself to the earth.}} {{font|size=smaller|color=#0000FF|53}}{{font|color=#000088| And the servant brought forth jewels of silver, and jewels of gold, and raiment, and gave them to Rebekah: he gave also to her brother and to her mother precious things.}} {{font|size=smaller|color=#0000FF|54}}{{font|color=#000088| And they did eat and drink, he and the men that were with him, and tarried all night; and they rose up in the morning, and he said, Send me away unto my master.}} {{font|size=smaller|color=#0000FF|55}}{{font|color=#000088| And her brother and her mother said, Let the damsel abide with us a few days, at the least ten; after that she shall go.}} {{font|size=smaller|color=#0000FF|56}}{{font|color=#000088| And he said unto them, Hinder me not, seeing the LORD hath prospered my way; send me away that I may go to my master.}} {{font|size=smaller|color=#0000FF|57}}{{font|color=#000088| And they said, We will call the damsel, and enquire at her mouth.}} {{font|size=smaller|color=#0000FF|58}}{{font|color=#000088| And they called Rebekah, and said unto her, Wilt thou go with this man? And she said, I will go.}} {{font|size=smaller|color=#0000FF|59}}{{font|color=#000088| And they sent away Rebekah their sister, and her nurse, and Abraham's servant, and his men.}} {{font|size=smaller|color=#0000FF|60}}{{font|color=#000088| And they blessed Rebekah, and said unto her, Thou art our sister, be thou the mother of thousands of millions, and let thy seed possess the gate of those which hate them.}} {{font|size=smaller|color=#0000FF|61}}{{font|color=#000088| And Rebekah arose, and her damsels, and they rode upon the camels, and followed the man: and the servant took Rebekah, and went his way.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1300061916233113600|title=Genesis 24:50-61|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|62}}{{font|color=#000088| And Isaac came from the way of the well Lahairoi; for he dwelt in the south country.}} {{font|size=smaller|color=#0000FF|63}}{{font|color=#000088| And Isaac went out to meditate in the field at the eventide: and he lifted up his eyes, and saw, and, behold, the camels were coming.}} {{font|size=smaller|color=#0000FF|64}}{{font|color=#000088| And Rebekah lifted up her eyes, and when she saw Isaac, she lighted off the camel.}} {{font|size=smaller|color=#0000FF|65}}{{font|color=#000088| For she had said unto the servant, What man is this that walketh in the field to meet us? And the servant had said, It is my master: therefore she took a vail, and covered herself.}} {{font|size=smaller|color=#0000FF|66}}{{font|color=#000088| And the servant told Isaac all things that he had done.}} {{font|size=smaller|color=#0000FF|67}}{{font|color=#000088| And Isaac brought her into}}<ref name=":21">{{Cite web|url=https://twitter.com/joelbaden/status/1300400768466460673|title=Genesis 24:62-67|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|color=#880000|his mother Sarah's}}<ref name=":21" /> {{font|color=#000088|tent, and took Rebekah, and she became his wife; and he loved her: and Isaac was comforted after his mother's death.}}<ref name=":21" /> ==Chapter 25== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Then again Abraham took a wife, and her name was Keturah.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And she bare him Zimran, and Jokshan, and Medan, and Midian, and Ishbak, and Shuah.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Jokshan begat Sheba, and Dedan. And the sons of Dedan were Asshurim, and Letushim, and Leummim.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And the sons of Midian; Ephah, and Epher, and Hanoch, and Abidah, and Eldaah. All these were the children of Keturah.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Abraham gave all that he had unto Isaac.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| But unto the sons of the concubines, which Abraham had, Abraham gave gifts, and sent them away from Isaac his son, while he yet lived, eastward, unto the east country.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1300770210262196224|title=Genesis 25:1-6|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And these are the days of the years of Abraham's life which he lived, an hundred threescore and fifteen years.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| Then Abraham gave up the ghost, and died}}<ref name=":22">{{Cite web|url=https://twitter.com/joelbaden/status/1301125810724761600|title=Genesis 25:7-11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-11}}</ref> {{font|color=#880000|in a good old age,}}<ref name=":22" /> {{font|color=#888800|an old man, and full of years; and was gathered to his people.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And his sons Isaac and Ishmael buried him in the cave of Machpelah, in the field of Ephron the son of Zohar the Hittite, which is before Mamre;}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| The field which Abraham purchased of the sons of Heth: there was Abraham buried, and Sarah his wife.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And it came to pass after the death of Abraham, that God blessed his son Isaac;}}<ref name=":22" /> {{font|color=#000088| and Isaac dwelt by the well Lahairoi.}}<ref name=":22" /> {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| Now these are the generations of Ishmael, Abraham's son, whom Hagar the Egyptian, Sarah's handmaid, bare unto Abraham:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And these are the names of the sons of Ishmael, by their names, according to their generations: the firstborn of Ishmael, Nebajoth; and Kedar, and Adbeel, and Mibsam,}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And Mishma, and Dumah, and Massa,}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| Hadar, and Tema, Jetur, Naphish, and Kedemah:}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| These are the sons of Ishmael, and these are their names, by their towns, and by their castles; twelve princes according to their nations.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And these are the years of the life of Ishmael, an hundred and thirty and seven years: and he gave up the ghost and died; and was gathered unto his people.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And they dwelt from Havilah unto Shur, that is before Egypt, as thou goest toward Assyria: and he died in the presence of all his brethren.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1301492506954207234|title=Genesis 25:12-18|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And these are the generations of Isaac, Abraham's son: Abraham begat Isaac:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And Isaac was forty years old when he took Rebekah to wife, the daughter of Bethuel the Syrian of Padanaram, the sister to Laban the Syrian.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1302225311229251584|title=Genesis 25:19-20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And Isaac intreated the LORD for his wife, because she was barren: and the LORD was intreated of him, and Rebekah his wife conceived.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And the children struggled together within her; and she said, If it be so, why am I thus? And she went to enquire of the LORD.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And the LORD said unto her, Two nations are in thy womb, and two manner of people shall be separated from thy bowels; and the one people shall be stronger than the other people; and the elder shall serve the younger.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And when her days to be delivered were fulfilled, behold, there were twins in her womb.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And the first came out red, all over like an hairy garment; and they called his name Esau.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And after that came his brother out, and his hand took hold on Esau's heel; and his name was called Jacob:}}<ref name=":24">{{Cite web|url=https://twitter.com/joelbaden/status/1302559271457099777|title=Genesis 25:21-26|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|color=#888800|and Isaac was threescore years old when she bare them.}}<ref name=":24" /> {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And the boys grew: and Esau was a cunning hunter, a man of the field; and Jacob was a plain man, dwelling in tents.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And Isaac loved Esau, because he did eat of his venison: but Rebekah loved Jacob.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And Jacob sod pottage: and Esau came from the field, and he was faint:}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And Esau said to Jacob, Feed me, I pray thee, with that same red pottage; for I am faint: therefore was his name called Edom.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And Jacob said, Sell me this day thy birthright.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And Esau said, Behold, I am at the point to die: and what profit shall this birthright do to me?}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And Jacob said, Swear to me this day; and he sware unto him: and he sold his birthright unto Jacob.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| Then Jacob gave Esau bread and pottage of lentiles; and he did eat and drink, and rose up, and went his way: thus Esau despised his birthright.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1302934923804921858|title=Genesis 25:27-34|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> ==Chapter 26== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And there was a famine in the land, beside the first famine that was in the days of Abraham. And Isaac went unto Abimelech king of the Philistines unto Gerar.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And the LORD appeared unto him, and said, Go not down into Egypt; dwell in the land which I shall tell thee of:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| Sojourn in this land, and I will be with thee, and will bless thee; for unto thee, and unto thy seed, I will give all these countries, and I will perform the oath which I sware unto Abraham thy father;}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And I will make thy seed to multiply as the stars of heaven, and will give unto thy seed all these countries; and in thy seed shall all the nations of the earth be blessed;}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| Because that Abraham obeyed my voice, and kept my charge, my commandments, my statutes, and my laws.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Isaac dwelt in Gerar:}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the men of the place asked him of his wife; and he said, She is my sister: for he feared to say, She is my wife; lest, said he, the men of the place should kill me for Rebekah; because she was fair to look upon.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And it came to pass, when he had been there a long time, that Abimelech king of the Philistines looked out at a window, and saw, and, behold, Isaac was sporting with Rebekah his wife.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Abimelech called Isaac, and said, Behold, of a surety she is thy wife: and how saidst thou, She is my sister? And Isaac said unto him, Because I said, Lest I die for her.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And Abimelech said, What is this thou hast done unto us? one of the people might lightly have lien with thy wife, and thou shouldest have brought guiltiness upon us.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And Abimelech charged all his people, saying, He that toucheth this man or his wife shall surely be put to death.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1304002657804324865|title=Genesis 26:1-11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| Then Isaac sowed in that land, and received in the same year an hundredfold: and the LORD blessed him.}}<ref name=":23" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And the man waxed great, and went forward, and grew until he became very great:}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| For he had possession of flocks, and possession of herds, and great store of servants: and the Philistines envied him.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| For all the wells which his father's servants had digged in the days of Abraham his father, the Philistines had stopped them, and filled them with earth.}}<ref name=":23" /> {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Abimelech said unto Isaac, Go from us; for thou art much mightier than we.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And Isaac departed thence, and pitched his tent in the valley of Gerar, and dwelt there.}}<ref name=":23" /> {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| And Isaac digged again the wells of water, which they had digged in the days of Abraham his father; for the Philistines had stopped them after the death of Abraham: and he called their names after the names by which his father had called them.}}<ref name=":23" /> {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And Isaac's servants digged in the valley, and found there a well of springing water.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And the herdmen of Gerar did strive with Isaac's herdmen, saying, The water is ours: and he called the name of the well Esek; because they strove with him.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And they digged another well, and strove for that also: and he called the name of it Sitnah.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And he removed from thence, and digged another well; and for that they strove not: and he called the name of it Rehoboth; and he said, For now the LORD hath made room for us, and we shall be fruitful in the land.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And he went up from thence to Beersheba.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And the LORD appeared unto him the same night, and said, I am the God of Abraham thy father: fear not, for I am with thee, and will bless thee, and multiply thy seed for my servant Abraham's sake.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And he builded an altar there, and called upon the name of the LORD, and pitched his tent there: and there Isaac's servants digged a well.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| Then Abimelech went to him from Gerar, and Ahuzzath one of his friends, and Phichol the chief captain of his army.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Isaac said unto them, Wherefore come ye to me, seeing ye hate me, and have sent me away from you?}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And they said, We saw certainly that the LORD was with thee: and we said, Let there be now an oath betwixt us, even betwixt us and thee, and let us make a covenant with thee;}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| That thou wilt do us no hurt, as we have not touched thee, and as we have done unto thee nothing but good, and have sent thee away in peace: thou art now the blessed of the LORD.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And he made them a feast, and they did eat and drink.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And they rose up betimes in the morning, and sware one to another: and Isaac sent them away, and they departed from him in peace.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And it came to pass the same day, that Isaac's servants came, and told him concerning the well which they had digged, and said unto him, We have found water.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And he called it Shebah: therefore the name of the city is Beersheba unto this day.}}<ref name=":23" /> {{font|size=smaller|color=#0000FF|34}}{{font|color=#888800| And Esau was forty years old when he took to wife Judith the daughter of Beeri the Hittite, and Bashemath the daughter of Elon the Hittite:}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#888800| Which were a grief of mind unto Isaac and to Rebekah.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1304750170287079426|title=Genesis 26:34-35|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> ==Chapter 27== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And it came to pass, that when Isaac was old, and his eyes were dim, so that he could not see, he called Esau his eldest son, and said unto him, My son: and he said unto him, Behold, here am I.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he said, Behold now, I am old, I know not the day of my death:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| Now therefore take, I pray thee, thy weapons, thy quiver and thy bow, and go out to the field, and take me some venison;}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And make me savoury meat, such as I love, and bring it to me, that I may eat; that my soul may bless thee before I die.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Rebekah heard when Isaac spake to Esau his son. And Esau went to the field to hunt for venison, and to bring it.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Rebekah spake unto Jacob her son, saying, Behold, I heard thy father speak unto Esau thy brother, saying,}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| Bring me venison, and make me savoury meat, that I may eat, and bless thee before the LORD before my death.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| Now therefore, my son, obey my voice according to that which I command thee.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| Go now to the flock, and fetch me from thence two good kids of the goats; and I will make them savoury meat for thy father, such as he loveth:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And thou shalt bring it to thy father, that he may eat, and that he may bless thee before his death.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And Jacob said to Rebekah his mother, Behold, Esau my brother is a hairy man, and I am a smooth man:}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| My father peradventure will feel me, and I shall seem to him as a deceiver; and I shall bring a curse upon me, and not a blessing.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And his mother said unto him, Upon me be thy curse, my son: only obey my voice, and go fetch me them.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And he went, and fetched, and brought them to his mother: and his mother made savoury meat, such as his father loved.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And Rebekah took goodly raiment of her eldest son Esau, which were with her in the house, and put them upon Jacob her younger son:}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And she put the skins of the kids of the goats upon his hands, and upon the smooth of his neck:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And she gave the savoury meat and the bread, which she had prepared, into the hand of her son Jacob.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1305116917146038272|title=Genesis 27:1-17|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And he came unto his father, and said, My father: and he said, Here am I; who art thou, my son?}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And Jacob said unto his father, I am Esau thy first born; I have done according as thou badest me: arise, I pray thee, sit and eat of my venison, that thy soul may bless me.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Isaac said unto his son, How is it that thou hast found it so quickly, my son? And he said, Because the LORD thy God brought it to me.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And Isaac said unto Jacob, Come near, I pray thee, that I may feel thee, my son, whether thou be my very son Esau or not.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Jacob went near unto Isaac his father; and he felt him, and said, The voice is Jacob's voice, but the hands are the hands of Esau.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And he discerned him not, because his hands were hairy, as his brother Esau's hands: so he blessed him.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And he said, Art thou my very son Esau? And he said, I am.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And he said, Bring it near to me, and I will eat of my son's venison, that my soul may bless thee. And he brought it near to him, and he did eat: and he brought him wine, and he drank.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And his father Isaac said unto him, Come near now, and kiss me, my son.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And he came near, and kissed him: and he smelled the smell of his raiment, and blessed him, and said, See, the smell of my son is as the smell of a field which the LORD hath blessed:}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| Therefore God give thee of the dew of heaven, and the fatness of the earth, and plenty of corn and wine:}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| Let people serve thee, and nations bow down to thee: be lord over thy brethren, and let thy mother's sons bow down to thee: cursed be every one that curseth thee, and blessed be he that blesseth thee.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1305471242472955904|title=Genesis 27:18-29|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And it came to pass, as soon as Isaac had made an end of blessing Jacob, and Jacob was yet scarce gone out from the presence of Isaac his father, that Esau his brother came in from his hunting.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he also had made savoury meat, and brought it unto his father, and said unto his father, Let my father arise, and eat of his son's venison, that thy soul may bless me.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And Isaac his father said unto him, Who art thou? And he said, I am thy son, thy firstborn Esau.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And Isaac trembled very exceedingly, and said, Who? where is he that hath taken venison, and brought it me, and I have eaten of all before thou camest, and have blessed him? yea, and he shall be blessed.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And when Esau heard the words of his father, he cried with a great and exceeding bitter cry, and said unto his father, Bless me, even me also, O my father.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And he said, Thy brother came with subtilty, and hath taken away thy blessing.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#000088| And he said, Is not he rightly named Jacob? for he hath supplanted me these two times: he took away my birthright; and, behold, now he hath taken away my blessing. And he said, Hast thou not reserved a blessing for me?}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#000088| And Isaac answered and said unto Esau, Behold, I have made him thy lord, and all his brethren have I given to him for servants; and with corn and wine have I sustained him: and what shall I do now unto thee, my son?}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#000088| And Esau said unto his father, Hast thou but one blessing, my father? bless me, even me also, O my father. And Esau lifted up his voice, and wept.}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#000088| And Isaac his father answered and said unto him, Behold, thy dwelling shall be the fatness of the earth, and of the dew of heaven from above;}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#000088| And by thy sword shalt thou live, and shalt serve thy brother; and it shall come to pass when thou shalt have the dominion, that thou shalt break his yoke from off thy neck.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1305830476754677765|title=Genesis 27:30-40|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|41}}{{font|color=#000088| And Esau hated Jacob because of the blessing wherewith his father blessed him: and Esau said in his heart, The days of mourning for my father are at hand; then will I slay my brother Jacob.}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#000088| And these words of Esau her elder son were told to Rebekah: and she sent and called Jacob her younger son, and said unto him, Behold, thy brother Esau, as touching thee, doth comfort himself, purposing to kill thee.}} {{font|size=smaller|color=#0000FF|43}}{{font|color=#000088| Now therefore, my son, obey my voice; and arise, flee thou to Laban my brother to Haran;}} {{font|size=smaller|color=#0000FF|44}}{{font|color=#000088| And tarry with him a few days, until thy brother's fury turn away;}} {{font|size=smaller|color=#0000FF|45}}{{font|color=#000088| Until thy brother's anger turn away from thee, and he forget that which thou hast done to him: then I will send, and fetch thee from thence: why should I be deprived also of you both in one day?}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1306205184499089409|title=Genesis 27:41-45|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> {{font|size=smaller|color=#0000FF|46}}{{font|color=#888800| And Rebekah said to Isaac, I am weary of my life because of the daughters of Heth: if Jacob take a wife of the daughters of Heth, such as these which are of the daughters of the land, what good shall my life do me?}}<ref name=":25">{{Cite web|url=https://twitter.com/joelbaden/status/1306553003336441856|title=Genesis 27:46-28:9|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-12}}</ref> ==Chapter 28== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And Isaac called Jacob, and blessed him, and charged him, and said unto him, Thou shalt not take a wife of the daughters of Canaan.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| Arise, go to Padanaram, to the house of Bethuel thy mother's father; and take thee a wife from thence of the daughters of Laban thy mother's brother.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And God Almighty bless thee, and make thee fruitful, and multiply thee, that thou mayest be a multitude of people;}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And give thee the blessing of Abraham, to thee, and to thy seed with thee; that thou mayest inherit the land wherein thou art a stranger, which God gave unto Abraham.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And Isaac sent away Jacob: and he went to Padanaram unto Laban, son of Bethuel the Syrian, the brother of Rebekah, Jacob's and Esau's mother.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| When Esau saw that Isaac had blessed Jacob, and sent him away to Padanaram, to take him a wife from thence; and that as he blessed him he gave him a charge, saying, Thou shalt not take a wife of the daughters of Canaan;}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And that Jacob obeyed his father and his mother, and was gone to Padanaram;}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And Esau seeing that the daughters of Canaan pleased not Isaac his father;}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| Then went Esau unto Ishmael, and took unto the wives which he had Mahalath the daughter of Ishmael Abraham's son, the sister of Nebajoth, to be his wife.}}<ref name=":25" /> {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And Jacob went out from Beersheba, and went toward Haran.}}<ref name=":26">{{Cite web|url=https://twitter.com/joelbaden/status/1306930592983003137|title=Genesis 28:10-22|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888|And he lighted upon a certain place, and tarried there all night, because the sun was set; and he took of the stones of that place, and put them for his pillows, and lay down in that place to sleep.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And he dreamed, and behold a ladder set up on the earth, and the top of it reached to heaven: and behold the angels of God ascending and descending on it.}}<ref name=":26" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And, behold, the LORD stood above it, and said, I am the LORD God of Abraham thy father, and the God of Isaac: the land whereon thou liest, to thee will I give it, and to thy seed;}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And thy seed shall be as the dust of the earth, and thou shalt spread abroad to the west, and to the east, and to the north, and to the south: and in thee and in thy seed shall all the families of the earth be blessed.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And, behold, I am with thee, and will keep thee in all places whither thou goest, and will bring thee again into this land; for I will not leave thee, until I have done that which I have spoken to thee of.}}<ref name=":26" /> {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| And Jacob awaked out of his sleep,}}<ref name=":26" /> {{font|color=#000088|and he said, Surely the LORD is in this place; and I knew it not.}}<ref name=":26" /> {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And he was afraid, and said, How dreadful is this place! this is none other but the house of God, and this is the gate of heaven.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Jacob rose up early in the morning, and took the stone that he had put for his pillows, and set it up for a pillar, and poured oil upon the top of it.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And he called the name of that place Bethel: but the name of that city was called Luz at the first.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888|And Jacob vowed a vow, saying,}}<ref name=":26" /> {{font|color=#000088|If God will be with me, and will keep me in this way that I go, and will give me bread to eat, and raiment to put on,}}<ref name=":26" /> {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| So that I come again to my father's house in peace;}}<ref name=":26" /> {{font|color=#000088|then shall the LORD be my God:}}<ref name=":26" /> {{font|size=smaller|color=#0000FF|22}}{{font|color=#880000| And}}<ref name=":26" /> {{font|color=#008888|this stone, which I have set for a pillar, shall be God's house: and of all that thou shalt give me I will surely give the tenth unto thee.}}<ref name=":26" /> ==Chapter 29== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Then Jacob went on his journey, and came into the land of the people of the east.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he looked, and behold a well in the field, and, lo, there were three flocks of sheep lying by it; for out of that well they watered the flocks: and a great stone was upon the well's mouth.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And thither were all the flocks gathered: and they rolled the stone from the well's mouth, and watered the sheep, and put the stone again upon the well's mouth in his place.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And Jacob said unto them, My brethren, whence be ye? And they said, Of Haran are we.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And he said unto them, Know ye Laban the son of Nahor? And they said, We know him.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And he said unto them, Is he well? And they said, He is well: and, behold, Rachel his daughter cometh with the sheep.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And he said, Lo, it is yet high day, neither is it time that the cattle should be gathered together: water ye the sheep, and go and feed them.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And they said, We cannot, until all the flocks be gathered together, and till they roll the stone from the well's mouth; then we water the sheep.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And while he yet spake with them, Rachel came with her father's sheep: for she kept them.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And it came to pass, when Jacob saw Rachel the daughter of Laban his mother's brother, and the sheep of Laban his mother's brother, that Jacob went near, and rolled the stone from the well's mouth, and watered the flock of Laban his mother's brother.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And Jacob kissed Rachel, and lifted up his voice, and wept.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And Jacob told Rachel that he was her father's brother, and that he was Rebekah's son: and she ran and told her father.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And it came to pass, when Laban heard the tidings of Jacob his sister's son, that he ran to meet him, and embraced him, and kissed him, and brought him to his house. And he told Laban all these things.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Laban said to him, Surely thou art my bone and my flesh. And he abode with him the space of a month.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1307275045224550400|title=Genesis 29:1-14|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And Laban said unto Jacob, Because thou art my brother, shouldest thou therefore serve me for nought? tell me, what shall thy wages be?}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Laban had two daughters: the name of the elder was Leah, and the name of the younger was Rachel.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| Leah was tender eyed; but Rachel was beautiful and well favoured.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And Jacob loved Rachel; and said, I will serve thee seven years for Rachel thy younger daughter.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And Laban said, It is better that I give her to thee, than that I should give her to another man: abide with me.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Jacob served seven years for Rachel; and they seemed unto him but a few days, for the love he had to her.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1307644185625931778|title=Genesis 29:15-20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And Jacob said unto Laban, Give me my wife, for my days are fulfilled, that I may go in unto her.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Laban gathered together all the men of the place, and made a feast.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And it came to pass in the evening, that he took Leah his daughter, and brought her to him; and he went in unto her.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#880000| And Laban gave unto his daughter Leah Zilpah his maid for an handmaid.}}<ref name=":27" /> {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And it came to pass, that in the morning, behold, it was Leah: and he said to Laban, What is this thou hast done unto me? did not I serve with thee for Rachel? wherefore then hast thou beguiled me?}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And Laban said, It must not be so done in our country, to give the younger before the firstborn.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| Fulfil her week, and we will give thee this also for the service which thou shalt serve with me yet seven other years.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And Jacob did so, and fulfilled her week: and he gave him Rachel his daughter to wife also.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#880000| And Laban gave to Rachel his daughter Bilhah his handmaid to be her maid.}}<ref name=":27" /> {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And he went in also unto Rachel, and he loved also Rachel more than Leah, and served with him yet seven other years.}}<ref name=":27">{{Cite web|url=https://twitter.com/joelbaden/status/1307992618052313088|title=Genesis 29:21-30|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And when the LORD saw that Leah was hated, he opened her womb: but Rachel was barren.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And Leah conceived, and bare a son, and she called his name Reuben: for she said, Surely the LORD hath looked upon my affliction; now therefore my husband will love me.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And she conceived again, and bare a son; and said, Because the LORD hath heard that I was hated, he hath therefore given me this son also: and she called his name Simeon.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And she conceived again, and bare a son; and said, Now this time will my husband be joined unto me, because I have born him three sons: therefore was his name called Levi.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And she conceived again, and bare a son: and she said, Now will I praise the LORD: therefore she called his name Judah; and left bearing.}}<ref name=":28">{{Cite web|url=https://twitter.com/joelbaden/status/1308356214418538496|title=Genesis 29:31-30:24|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> ==Chapter 30== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And when Rachel saw that she bare Jacob no children, Rachel envied her sister; and said unto Jacob, Give me children, or else I die.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Jacob's anger was kindled against Rachel: and he said, Am I in God's stead, who hath withheld from thee the fruit of the womb?}}<ref name=":28" /> {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And she said, Behold my maid Bilhah, go in unto her; and she shall bear upon my knees, that I may also have children by her.}}<ref name=":28" /> {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And she gave him Bilhah her handmaid to wife:}}<ref name=":28" /> {{font|color=#008888| and Jacob went in unto her.}}<ref name=":28" /> {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Bilhah conceived, and bare Jacob a son.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Rachel said, God hath judged me, and hath also heard my voice, and hath given me a son: therefore called she his name Dan.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Bilhah Rachel's maid conceived again, and bare Jacob a second son.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Rachel said, With great wrestlings have I wrestled with my sister, and I have prevailed: and she called his name Naphtali.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| When Leah saw that she had left bearing, she took Zilpah her maid, and gave her Jacob to wife.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And Zilpah Leah's maid bare Jacob a son.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And Leah said, A troop cometh: and she called his name Gad.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And Zilpah Leah's maid bare Jacob a second son.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And Leah said, Happy am I, for the daughters will call me blessed: and she called his name Asher.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Reuben went in the days of wheat harvest, and found mandrakes in the field, and brought them unto his mother Leah. Then Rachel said to Leah, Give me, I pray thee, of thy son's mandrakes.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And she said unto her, Is it a small matter that thou hast taken my husband? and wouldest thou take away my son's mandrakes also? And Rachel said, Therefore he shall lie with thee to night for thy son's mandrakes.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Jacob came out of the field in the evening, and Leah went out to meet him, and said, Thou must come in unto me; for surely I have hired thee with my son's mandrakes. And he lay with her that night.}}<ref name=":28" /> {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And God hearkened unto Leah, and she conceived, and bare Jacob the fifth son.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Leah said, God hath given me my hire, because I have given my maiden to my husband: and she called his name Issachar.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And Leah conceived again, and bare Jacob the sixth son.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And Leah said, God hath endued me with a good dowry;}} {{font|color=#000088|now will my husband dwell with me, because I have born him six sons:}} {{font|color=#008888|and she called his name Zebulun.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And afterwards she bare a daughter, and called her name Dinah.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| And God remembered Rachel, and God hearkened to her, and opened her womb.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| And she conceived, and bare a son; and said, God hath taken away my reproach:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And she called his name Joseph;}}<ref name=":28" /> {{font|color=#000088| and said, The LORD shall add to me another son.}}<ref name=":28" /> {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And it came to pass, when Rachel had born Joseph, that Jacob said unto Laban, Send me away, that I may go unto mine own place, and to my country.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| Give me my wives and my children, for whom I have served thee, and let me go: for thou knowest my service which I have done thee.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And Laban said unto him, I pray thee, if I have found favour in thine eyes, tarry: for I have learned by experience that the LORD hath blessed me for thy sake.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And he said, Appoint me thy wages, and I will give it.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And he said unto him, Thou knowest how I have served thee, and how thy cattle was with me.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| For it was little which thou hadst before I came, and it is now increased unto a multitude; and the LORD hath blessed thee since my coming: and now when shall I provide for mine own house also?}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he said, What shall I give thee? And Jacob said, Thou shalt not give me any thing: if thou wilt do this thing for me, I will again feed and keep thy flock.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| I will pass through all thy flock to day, removing from thence all the speckled and spotted cattle, and all the brown cattle among the sheep, and the spotted and speckled among the goats: and of such shall be my hire.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| So shall my righteousness answer for me in time to come, when it shall come for my hire before thy face: every one that is not speckled and spotted among the goats, and brown among the sheep, that shall be counted stolen with me.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And Laban said, Behold, I would it might be according to thy word.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And he removed that day the he goats that were ringstraked and spotted, and all the she goats that were speckled and spotted, and every one that had some white in it, and all the brown among the sheep, and gave them into the hand of his sons.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#000088| And he set three days' journey betwixt himself and Jacob: and Jacob fed the rest of Laban's flocks.}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#000088| And Jacob took him rods of green poplar, and of the hazel and chestnut tree; and pilled white strakes in them, and made the white appear which was in the rods.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#000088| And he set the rods which he had pilled before the flocks in the gutters in the watering troughs when the flocks came to drink, that they should conceive when they came to drink.}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#000088| And the flocks conceived before the rods, and brought forth cattle ringstraked, speckled, and spotted.}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#000088| And Jacob did separate the lambs, and set the faces of the flocks toward the ringstraked, and all the brown in the flock of Laban; and he put his own flocks by themselves, and put them not unto Laban's cattle.}} {{font|size=smaller|color=#0000FF|41}}{{font|color=#000088| And it came to pass, whensoever the stronger cattle did conceive, that Jacob laid the rods before the eyes of the cattle in the gutters, that they might conceive among the rods.}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#000088| But when the cattle were feeble, he put them not in: so the feebler were Laban's, and the stronger Jacob's.}}<ref name=":29">{{Cite web|url=https://twitter.com/joelbaden/status/1308738228137193473|title=Genesis 30:25-43|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|43}}{{font|color=#888800| And the man increased exceedingly, and had much cattle, and maidservants, and menservants, and camels, and asses.}}<ref name=":29" /> ==Chapter 31== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And he heard the words of Laban's sons, saying, Jacob hath taken away all that was our father's; and of that which was our father's hath he gotten all this glory.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And Jacob beheld the countenance of Laban, and, behold, it was not toward him as before.}}<ref name=":30">{{Cite web|url=https://twitter.com/joelbaden/status/1309083507566809094|title=Genesis 31:1-17|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And the LORD said unto Jacob, Return unto the land of thy fathers, and to thy kindred; and I will be with thee.}}<ref name=":30" /> {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| And Jacob sent and called Rachel and Leah to the field unto his flock,}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And said unto them, I see your father's countenance, that it is not toward me as before; but the God of my father hath been with me.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And ye know that with all my power I have served your father.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| And your father hath deceived me, and changed my wages ten times; but God suffered him not to hurt me.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| If he said thus, The speckled shall be thy wages; then all the cattle bare speckled: and if he said thus, The ringstraked shall be thy hire; then bare all the cattle ringstraked.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| Thus God hath taken away the cattle of your father, and given them to me.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And it came to pass at the time that the cattle conceived, that I lifted up mine eyes, and saw in a dream, and, behold, the rams which leaped upon the cattle were ringstraked, speckled, and grisled.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And the angel of God spake unto me in a dream, saying, Jacob: And I said, Here am I.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And he said, Lift up now thine eyes, and see, all the rams which leap upon the cattle are ringstraked, speckled, and grisled: for I have seen all that Laban doeth unto thee.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| I am the God of Bethel, where thou anointedst the pillar, and where thou vowedst a vow unto me: now arise, get thee out from this land, and return unto the land of thy kindred.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And Rachel and Leah answered and said unto him, Is there yet any portion or inheritance for us in our father's house?}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| Are we not counted of him strangers? for he hath sold us, and hath quite devoured also our money.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| For all the riches which God hath taken from our father, that is ours, and our children's: now then, whatsoever God hath said unto thee, do.}}<ref name=":30" /> {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| Then Jacob rose up, and set his sons and his wives upon camels;}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800|And he carried away all his cattle, and all his goods which he had gotten, the cattle of his getting, which he had gotten in Padanaram, for to go to Isaac his father in the land of Canaan.}}<ref name=":30" /> {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And Laban went to shear his sheep: and Rachel had stolen the images that were her father's.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And Jacob stole away unawares to Laban the Syrian, in that he told him not that he fled.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| So he fled with all that he had; and he rose up, and passed over the river, and set his face toward the mount Gilead.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| And it was told Laban on the third day that Jacob was fled.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| And he took his brethren with him, and pursued after him seven days' journey; and they overtook him in the mount Gilead.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And God came to Laban the Syrian in a dream by night, and said unto him, Take heed that thou speak not to Jacob either good or bad.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| Then Laban overtook Jacob. Now Jacob had pitched his tent in the mount: and Laban with his brethren pitched in the mount of Gilead.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#008888| And Laban said to Jacob, What hast thou done, that thou hast stolen away unawares to me, and carried away my daughters, as captives taken with the sword?}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#008888| Wherefore didst thou flee away secretly, and steal away from me; and didst not tell me, that I might have sent thee away with mirth, and with songs, with tabret, and with harp?}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| And hast not suffered me to kiss my sons and my daughters? thou hast now done foolishly in so doing.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#008888| It is in the power of my hand to do you hurt: but the God of your father spake unto me yesternight, saying, Take thou heed that thou speak not to Jacob either good or bad.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#008888| And now, though thou wouldest needs be gone, because thou sore longedst after thy father's house, yet wherefore hast thou stolen my gods?}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#008888| And Jacob answered and said to Laban, Because I was afraid: for I said, Peradventure thou wouldest take by force thy daughters from me.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#008888| With whomsoever thou findest thy gods, let him not live: before our brethren discern thou what is thine with me, and take it to thee. For Jacob knew not that Rachel had stolen them.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#008888| And Laban went into Jacob's tent, and into Leah's tent, and into the two maidservants' tents; but he found them not. Then went he out of Leah's tent, and entered into Rachel's tent.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#008888| Now Rachel had taken the images, and put them in the camel's furniture, and sat upon them. And Laban searched all the tent, but found them not.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#008888| And she said to her father, Let it not displease my lord that I cannot rise up before thee; for the custom of women is upon me. And he searched, but found not the images.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#008888| And Jacob was wroth, and chode with Laban: and Jacob answered and said to Laban, What is my trespass? what is my sin, that thou hast so hotly pursued after me?}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#008888| Whereas thou hast searched all my stuff, what hast thou found of all thy household stuff? set it here before my brethren and thy brethren, that they may judge betwixt us both.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#008888| This twenty years have I been with thee; thy ewes and thy she goats have not cast their young, and the rams of thy flock have I not eaten.}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#008888| That which was torn of beasts I brought not unto thee; I bare the loss of it; of my hand didst thou require it, whether stolen by day, or stolen by night.}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#008888| Thus I was; in the day the drought consumed me, and the frost by night; and my sleep departed from mine eyes.}} {{font|size=smaller|color=#0000FF|41}}{{font|color=#008888| Thus have I been twenty years in thy house; I served thee fourteen years for thy two daughters, and six years for thy cattle: and thou hast changed my wages ten times.}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#008888| Except the God of my father, the God of Abraham, and the fear of Isaac, had been with me, surely thou hadst sent me away now empty. God hath seen mine affliction and the labour of my hands, and rebuked thee yesternight.}} {{font|size=smaller|color=#0000FF|43}}{{font|color=#008888| And Laban answered and said unto Jacob, These daughters are my daughters, and these children are my children, and these cattle are my cattle, and all that thou seest is mine: and what can I do this day unto these my daughters, or unto their children which they have born?}} {{font|size=smaller|color=#0000FF|44}}{{font|color=#008888| Now therefore come thou, let us make a covenant, I and thou; and let it be for a witness between me and thee.}} {{font|size=smaller|color=#0000FF|45}}{{font|color=#008888| And Jacob took a stone, and set it up for a pillar.}} {{font|size=smaller|color=#0000FF|46}}{{font|color=#008888| And Jacob said unto his brethren, Gather stones; and they took stones, and made an heap: and they did eat there upon the heap.}} {{font|size=smaller|color=#0000FF|47}}{{font|color=#008888| And Laban called it Jegarsahadutha: but Jacob called it Galeed.}} {{font|size=smaller|color=#0000FF|48}}{{font|color=#008888| And Laban said, This heap is a witness between me and thee this day. Therefore was the name of it called Galeed;}}<ref name=":30" /> {{font|size=smaller|color=#0000FF|49}}{{font|color=#880000| And Mizpah; for he said, The LORD watch between me and thee, when we are absent one from another.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1309453674943479808|title=Genesis 31:19-54|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-13}}</ref> {{font|size=smaller|color=#0000FF|50}}{{font|color=#008888| If thou shalt afflict my daughters, or if thou shalt take other wives beside my daughters, no man is with us; see, God is witness betwixt me and thee.}} {{font|size=smaller|color=#0000FF|51}}{{font|color=#008888| And Laban said to Jacob, Behold this heap, and behold this pillar, which I have cast betwixt me and thee:}} {{font|size=smaller|color=#0000FF|52}}{{font|color=#008888| This heap be witness, and this pillar be witness, that I will not pass over this heap to thee, and that thou shalt not pass over this heap and this pillar unto me, for harm.}} {{font|size=smaller|color=#0000FF|53}}{{font|color=#008888| The God of Abraham, and the God of Nahor, the God of their father, judge betwixt us. And Jacob sware by the fear of his father Isaac.}} {{font|size=smaller|color=#0000FF|54}}{{font|color=#008888| Then Jacob offered sacrifice upon the mount, and called his brethren to eat bread: and they did eat bread, and tarried all night in the mount.}}<ref name=":30" /> {{font|size=smaller|color=#0000FF|55}}{{font|color=#008888| And early in the morning Laban rose up, and kissed his sons and his daughters, and blessed them: and Laban departed, and returned unto his place.}}<ref name=":31" /> ==Chapter 32== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And Jacob went on his way, and the angels of God met him.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And when Jacob saw them, he said, This is God's host:}} {{font|color=#880000| and he called the name of that place Mahanaim.}}<ref name=":31">{{Cite web|url=https://twitter.com/joelbaden/status/1309809807340244992|title=Genesis 32:1-3|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Jacob sent messengers before him to Esau his brother unto the land of Seir, the country of Edom.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And he commanded them, saying, Thus shall ye speak unto my lord Esau; Thy servant Jacob saith thus, I have sojourned with Laban, and stayed there until now:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And I have oxen, and asses, flocks, and menservants, and womenservants: and I have sent to tell my lord, that I may find grace in thy sight.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And the messengers returned to Jacob, saying, We came to thy brother Esau, and also he cometh to meet thee, and four hundred men with him.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| Then Jacob was greatly afraid and distressed: and he divided the people that was with him, and the flocks, and herds, and the camels, into two bands;}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And said, If Esau come to the one company, and smite it, then the other company which is left shall escape.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Jacob said, O God of my father Abraham, and God of my father Isaac, the LORD which saidst unto me, Return unto thy country, and to thy kindred, and I will deal well with thee:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| I am not worthy of the least of all the mercies, and of all the truth, which thou hast shewed unto thy servant; for with my staff I passed over this Jordan; and now I am become two bands.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Deliver me, I pray thee, from the hand of my brother, from the hand of Esau: for I fear him, lest he will come and smite me, and the mother with the children.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And thou saidst, I will surely do thee good, and make thy seed as the sand of the sea, which cannot be numbered for multitude.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And he lodged there that same night; and took of that which came to his hand a present for Esau his brother;}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Two hundred she goats, and twenty he goats, two hundred ewes, and twenty rams,}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| Thirty milch camels with their colts, forty kine, and ten bulls, twenty she asses, and ten foals.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And he delivered them into the hand of his servants, every drove by themselves; and said unto his servants, Pass over before me, and put a space betwixt drove and drove.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And he commanded the foremost, saying, When Esau my brother meeteth thee, and asketh thee, saying, Whose art thou? and whither goest thou? and whose are these before thee?}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| Then thou shalt say, They be thy servant Jacob's; it is a present sent unto my lord Esau: and, behold, also he is behind us.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And so commanded he the second, and the third, and all that followed the droves, saying, On this manner shall ye speak unto Esau, when ye find him.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And say ye moreover, Behold, thy servant Jacob is behind us. For he said, I will appease him with the present that goeth before me, and afterward I will see his face; peradventure he will accept of me.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| So went the present over before him: and himself lodged that night in the company.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And he rose up that night, and took his two wives, and his two womenservants, and his eleven sons, and passed over the ford Jabbok.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And he took them, and sent them over the brook, and sent over that he had.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1310164809741209601|title=Genesis 32:4-24|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And Jacob was left alone; and there wrestled a man with him until the breaking of the day.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And when he saw that he prevailed not against him, he touched the hollow of his thigh; and the hollow of Jacob's thigh was out of joint, as he wrestled with him.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And he said, Let me go, for the day breaketh. And he said, I will not let thee go, except thou bless me.}}<ref name=":32">{{Cite web|url=https://twitter.com/joelbaden/status/1310910760743206917|title=Genesis 32:25-33|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|27}}{{font|color=#008888| And he said unto him, What is thy name? And he said, Jacob.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God}}<ref name=":32" /> {{font|color=#880000|and with men, and hast prevailed.}}<ref name=":32" /> {{font|size=smaller|color=#0000FF|29}} {{font|color=#008888| And Jacob asked him, and said, Tell me, I pray thee, thy name. And he said, Wherefore is it that thou dost ask after my name?}}<ref name=":32" /> {{font|color=#000088|And he blessed him there.}}<ref name=":32" /> {{font|size=smaller|color=#0000FF|30}}{{font|color=#008888| And Jacob called the name of the place Peniel: for I have seen God face to face, and my life is preserved.}}<ref name=":32" /> {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And}}<ref name=":32" /> {{font|color=#880000|as he passed over Penuel}}<ref name=":32" /> {{font|color=#000088|the sun rose upon him, and he halted upon his thigh.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| Therefore the children of Israel eat not of the sinew which shrank, which is upon the hollow of the thigh, unto this day: because he touched the hollow of Jacob's thigh in the sinew that shrank.}}<ref name=":32" /> ==Chapter 33== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Jacob lifted up his eyes, and looked, and, behold, Esau came, and with him four hundred men. And he divided the children unto Leah, and unto Rachel, and unto the two handmaids.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he put the handmaids and their children foremost, and Leah and her children after, and Rachel and Joseph hindermost.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And he passed over before them, and bowed himself to the ground seven times, until he came near to his brother.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And Esau ran to meet him, and embraced him, and fell on his neck, and kissed him: and they wept.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And he lifted up his eyes, and saw the women and the children; and said, Who are those with thee? And he said, The children which God hath graciously given thy servant.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| Then the handmaidens came near, they and their children, and they bowed themselves.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Leah also with her children came near, and bowed themselves: and after came Joseph near and Rachel, and they bowed themselves.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And he said, What meanest thou by all this drove which I met? And he said, These are to find grace in the sight of my lord.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Esau said, I have enough, my brother; keep that thou hast unto thyself.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And Jacob said, Nay, I pray thee, if now I have found grace in thy sight, then receive my present at my hand: for therefore I have seen thy face, as though I had seen the face of God, and thou wast pleased with me.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Take, I pray thee, my blessing that is brought to thee; because God hath dealt graciously with me, and because I have enough. And he urged him, and he took it.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he said, Let us take our journey, and let us go, and I will go before thee.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And he said unto him, My lord knoweth that the children are tender, and the flocks and herds with young are with me: and if men should overdrive them one day, all the flock will die.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Let my lord, I pray thee, pass over before his servant: and I will lead on softly, according as the cattle that goeth before me and the children be able to endure, until I come unto my lord unto Seir.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And Esau said, Let me now leave with thee some of the folk that are with me. And he said, What needeth it? Let me find grace in the sight of my lord.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| So Esau returned that day on his way unto Seir.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1311258919218667520|title=Genesis 33:1-16|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And Jacob journeyed to Succoth, and built him an house, and made booths for his cattle: therefore the name of the place is called Succoth.}}<ref name=":33">{{Cite web|url=https://twitter.com/joelbaden/status/1311639589862416387|title=Genesis 33:17-20|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Jacob came to Shalem, a city of Shechem, }}<ref name=":33" /> {{font|color=#888800| which is in the land of Canaan, when he came from Padanaram;}}<ref name=":33" /> {{font|color=#008888| and pitched his tent before the city}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And he bought a parcel of a field, where he had spread his tent, at the hand of the children of Hamor, Shechem's father, for an hundred pieces of money.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And he erected there an altar, and called it EleloheIsrael.}}<ref name=":33" /> ==Chapter 34== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Dinah the daughter of Leah, which she bare unto Jacob, went out to see the daughters of the land.}}<ref name=":34">{{Cite web|url=https://twitter.com/joelbaden/status/1312003631491813378|title=Genesis 34|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-14}}</ref> {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And when Shechem the son of Hamor the Hivite, prince of the country, saw her,}}<ref name=":34" /> {{font|color=#000088|he took her, and lay with her, and defiled her.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And his soul clave unto Dinah the daughter of Jacob, and he loved the damsel, and spake kindly unto the damsel.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| And Shechem spake unto his father Hamor, saying, Get me this damsel to wife.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Jacob heard that he had defiled Dinah his daughter: now his sons were with his cattle in the field: and Jacob held his peace until they were come.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And Hamor the father of Shechem went out unto Jacob to commune with him.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And the sons of Jacob came out of the field when they heard it: and the men were grieved, and they were very wroth, because he had wrought folly in Israel in lying with Jacob's daughter; which thing ought not to be done.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And Hamor communed with them, saying, The soul of my son Shechem longeth for your daughter: I pray you give her him to wife.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And make ye marriages with us, and give your daughters unto us, and take our daughters unto you.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And ye shall dwell with us: and the land shall be before you; dwell and trade ye therein, and get you possessions therein.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And Shechem said unto her father and unto her brethren, Let me find grace in your eyes, and what ye shall say unto me I will give.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| Ask me never so much dowry and gift, and I will give according as ye shall say unto me: but give me the damsel to wife.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And the sons of Jacob answered Shechem and Hamor his father deceitfully,}}<ref name=":34" /> {{font|color=#880000|because he had defiled Dinah their sister:}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And they said unto them, We cannot do this thing, to give our sister to one that is uncircumcised; for that were a reproach unto us:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| But in this will we consent unto you: If ye will be as we be, that every male of you be circumcised;}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| Then will we give our daughters unto you, and we will take your daughters to us, and we will dwell with you, and we will become one people.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| But if ye will not hearken unto us, to be circumcised; then will we take our daughter, and we will be gone.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And their words pleased Hamor, and Shechem Hamor's son.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And the young man deferred not to do the thing, because he had delight in Jacob's daughter: and he was more honourable than all the house of his father.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And Hamor and Shechem his son came unto the gate of their city, and communed with the men of their city, saying,}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| These men are peaceable with us; therefore let them dwell in the land, and trade therein; for the land, behold, it is large enough for them; let us take their daughters to us for wives, and let us give them our daughters.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| Only herein will the men consent unto us for to dwell with us, to be one people, if every male among us be circumcised, as they are circumcised.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| Shall not their cattle and their substance and every beast of theirs be ours? only let us consent unto them, and they will dwell with us.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And unto Hamor and unto Shechem his son hearkened all that went out of the gate of his city; and every male was circumcised, all that went out of the gate of his city.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| And it came to pass on the third day, when they were sore, }}<ref name=":34" /> {{font|color=#000088|that two of the sons of Jacob, Simeon and Levi, Dinah's brethren, took each man his sword,}}<ref name=":34" /> {{font|color=#008888|and came upon the city boldly, and slew all the males.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And they slew Hamor and Shechem his son with the edge of the sword, and took Dinah out of Shechem's house, and went out.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|27}}{{font|color=#008888| The sons of Jacob came upon the slain, and spoiled the city,}}<ref name=":34" /> {{font|color=#880000|because they had defiled their sister.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| They took their sheep, and their oxen, and their asses, and that which was in the city, and that which was in the field,}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#008888| And all their wealth, and all their little ones, and their wives took they captive, and spoiled even all that was in the house.}}<ref name=":34" /> {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And Jacob said to Simeon and Levi, Ye have troubled me to make me to stink among the inhabitants of the land, among the Canaanites and the Perizzites: and I being few in number, they shall gather themselves together against me, and slay me; and I shall be destroyed, I and my house.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And they said, Should he deal with our sister as with an harlot?}}<ref name=":34" /> ==Chapter 35== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And God said unto Jacob, Arise, go up to Bethel, and dwell there: and make there an altar unto God, that appeared unto thee when thou fleddest from the face of Esau thy brother.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| Then Jacob said unto his household, and to all that were with him, Put away the strange gods that are among you, and be clean, and change your garments:}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And let us arise, and go up to Bethel; and I will make there an altar unto God, who answered me in the day of my distress, and was with me in the way which I went.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And they gave unto Jacob all the strange gods which were in their hand, and all their earrings which were in their ears; and Jacob hid them under the oak which was by Shechem.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And they journeyed: and the terror of God was upon the cities that were round about them, and they did not pursue after the sons of Jacob.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1312367856378675200|title=Genesis 35:1-5|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| So Jacob came to Luz,}}<ref name=":35">{{Cite web|url=https://twitter.com/joelbaden/status/1312701630328995841|title=Genesis 35:6-8|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000| which is in the land of Canaan, that is,}}<ref name=":35" /> {{font|color=#000088|Bethel, he and all the people that were with him.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And he built there an altar, and called the place Elbethel: because there God appeared unto him, when he fled from the face of his brother.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| But Deborah Rebekah's nurse died, and she was buried beneath Bethel under an oak: and the name of it was called Allonbachuth.}}<ref name=":35" /> {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And God appeared unto Jacob again, when he came out of Padanaram, and blessed him.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And God said unto him, Thy name is Jacob: thy name shall not be called any more Jacob, but Israel shall be thy name: and he called his name Israel.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And God said unto him, I am God Almighty: be fruitful and multiply; a nation and a company of nations shall be of thee, and kings shall come out of thy loins;}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And the land which I gave Abraham and Isaac, to thee I will give it, and to thy seed after thee will I give the land.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And God went up from him}}<ref name=":36">{{Cite web|url=https://twitter.com/joelbaden/status/1313080951044440064|title=Genesis 35:9-15|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|in the place where he talked with him.}}<ref name=":36" /> {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Jacob set up a pillar}}<ref name=":36" /> {{font|color=#880000|in the place where he talked with him,}}<ref name=":36" /> {{font|color=#000088|even a pillar of stone: and he poured a drink offering thereon, and he poured oil thereon.}}<ref name=":36" /> {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| And Jacob called the name of the place where God spake with him, Bethel.}}<ref name=":36" /> {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And they journeyed from Bethel; and there was but a little way to come to Ephrath:}}<ref name=":37">{{Cite web|url=https://twitter.com/joelbaden/status/1313448457273856006|title=Genesis 35:16-26|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#008888|and Rachel travailed, and she had hard labour.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And it came to pass, when she was in hard labour, that the midwife said unto her, Fear not; thou shalt have this son also.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And it came to pass, as her soul was in departing, (for she died) that she called his name Benoni: but his father called him Benjamin.}}<ref name=":37" /> {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| And Rachel died, and was buried in the way to Ephrath, which is Bethlehem.}}<ref name=":37" /> {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And Jacob set a pillar upon her grave: that is the pillar of Rachel's grave unto this day.}}<ref name=":37" /> {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And Israel journeyed, and spread his tent beyond the tower of Edar.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And it came to pass, when Israel dwelt in that land, that Reuben went and lay with Bilhah his father's concubine: and Israel heard it. }}<ref name=":37" /> {{font|color=#888800|Now the sons of Jacob were twelve:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| The sons of Leah; Reuben, Jacob's firstborn, and Simeon, and Levi, and Judah, and Issachar, and Zebulun:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| The sons of Rachel; Joseph, and Benjamin:}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| And the sons of Bilhah, Rachel's handmaid; Dan, and Naphtali:}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| And the sons of Zilpah, Leah's handmaid; Gad, and Asher: these are the sons of Jacob, which were born to him in Padanaram.}}<ref name=":37" /> {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| And Jacob came unto Isaac his father unto Mamre, unto the city of Arbah, which is Hebron, where Abraham and Isaac sojourned.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And the days of Isaac were an hundred and fourscore years.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#888800| And Isaac gave up the ghost, and died, and was gathered unto his people, being old and full of days: and his sons Esau and Jacob buried him.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1313814664518733824|title=Genesis 35:27-29|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> ==Chapter 36== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| Now these are the generations of Esau, who is Edom.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| Esau took his wives of the daughters of Canaan; Adah the daughter of Elon the Hittite, and Aholibamah the daughter of Anah the daughter of Zibeon the Hivite;}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Bashemath Ishmael's daughter, sister of Nebajoth.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And Adah bare to Esau Eliphaz; and Bashemath bare Reuel;}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And Aholibamah bare Jeush, and Jaalam, and Korah: these are the sons of Esau, which were born unto him in the land of Canaan.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And Esau took his wives, and his sons, and his daughters, and all the persons of his house, and his cattle, and all his beasts, and all his substance, which he had got in the land of Canaan; and went into the country from the face of his brother Jacob.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| For their riches were more than that they might dwell together; and the land wherein they were strangers could not bear them because of their cattle.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| Thus dwelt Esau in mount Seir: Esau is Edom.}}<ref name=":38">{{Cite web|url=https://twitter.com/joelbaden/status/1314171142282702848|title=Genesis 36|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|9}}{{font|color=#880000| And these are the generations of Esau the father of the Edomites in mount Seir:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#880000| These are the names of Esau's sons; Eliphaz the son of Adah the wife of Esau, Reuel the son of Bashemath the wife of Esau.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#880000| And the sons of Eliphaz were Teman, Omar, Zepho, and Gatam, and Kenaz.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#880000| And Timna was concubine to Eliphaz Esau's son; and she bare to Eliphaz Amalek: these were the sons of Adah Esau's wife.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#880000| And these are the sons of Reuel; Nahath, and Zerah, Shammah, and Mizzah: these were the sons of Bashemath Esau's wife.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#880000| And these were the sons of Aholibamah, the daughter of Anah the daughter of Zibeon, Esau's wife: and she bare to Esau Jeush, and Jaalam, and Korah.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#880000| These were dukes of the sons of Esau: the sons of Eliphaz the firstborn son of Esau; duke Teman, duke Omar, duke Zepho, duke Kenaz,}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#880000| Duke Korah, duke Gatam, and duke Amalek: these are the dukes that came of Eliphaz in the land of Edom; these were the sons of Adah.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#880000| And these are the sons of Reuel Esau's son; duke Nahath, duke Zerah, duke Shammah, duke Mizzah: these are the dukes that came of Reuel in the land of Edom; these are the sons of Bashemath Esau's wife.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#880000| And these are the sons of Aholibamah Esau's wife; duke Jeush, duke Jaalam, duke Korah: these were the dukes that came of Aholibamah the daughter of Anah, Esau's wife.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#880000| These are the sons of Esau, who is Edom, and these are their dukes.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#880000| These are the sons of Seir the Horite, who inhabited the land; Lotan, and Shobal, and Zibeon, and Anah,}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#880000| And Dishon, and Ezer, and Dishan: these are the dukes of the Horites, the children of Seir in the land of Edom.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#880000| And the children of Lotan were Hori and Hemam; and Lotan's sister was Timna.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#880000| And the children of Shobal were these; Alvan, and Manahath, and Ebal, Shepho, and Onam.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#880000| And these are the children of Zibeon; both Ajah, and Anah: this was that Anah that found the mules in the wilderness, as he fed the asses of Zibeon his father.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#880000| And the children of Anah were these; Dishon, and Aholibamah the daughter of Anah.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#880000| And these are the children of Dishon; Hemdan, and Eshban, and Ithran, and Cheran.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#880000| The children of Ezer are these; Bilhan, and Zaavan, and Akan.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#880000| The children of Dishan are these; Uz, and Aran.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#880000| These are the dukes that came of the Horites; duke Lotan, duke Shobal, duke Zibeon, duke Anah,}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#880000| Duke Dishon, duke Ezer, duke Dishan: these are the dukes that came of Hori, among their dukes in the land of Seir.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#880000| And these are the kings that reigned in the land of Edom, before there reigned any king over the children of Israel.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#880000| And Bela the son of Beor reigned in Edom: and the name of his city was Dinhabah.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#880000| And Bela died, and Jobab the son of Zerah of Bozrah reigned in his stead.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#880000| And Jobab died, and Husham of the land of Temani reigned in his stead.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#880000| And Husham died, and Hadad the son of Bedad, who smote Midian in the field of Moab, reigned in his stead: and the name of his city was Avith.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#880000| And Hadad died, and Samlah of Masrekah reigned in his stead.}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#880000| And Samlah died, and Saul of Rehoboth by the river reigned in his stead.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#880000| And Saul died, and Baalhanan the son of Achbor reigned in his stead.}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#880000| And Baalhanan the son of Achbor died, and Hadar reigned in his stead: and the name of his city was Pau; and his wife's name was Mehetabel, the daughter of Matred, the daughter of Mezahab.}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#880000| And these are the names of the dukes that came of Esau, according to their families, after their places, by their names; duke Timnah, duke Alvah, duke Jetheth,}} {{font|size=smaller|color=#0000FF|41}}{{font|color=#880000| Duke Aholibamah, duke Elah, duke Pinon,}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#880000| Duke Kenaz, duke Teman, duke Mibzar,}} {{font|size=smaller|color=#0000FF|43}}{{font|color=#880000| Duke Magdiel, duke Iram: these be the dukes of Edom, according to their habitations in the land of their possession: he is Esau the father of the Edomites.}}<ref name=":38" /> ==Chapter 37== {{font|size=smaller|color=#0000FF|1}}{{font|color=#888800| And Jacob dwelt in the land wherein his father was a stranger, in the land of Canaan.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#888800| These are the generations of Jacob.}}<ref name=":39">{{Cite web|url=https://twitter.com/joelbaden/status/1314910132288053249|title=Genesis 37:1-11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#008888| Joseph, being seventeen years old, was feeding the flock with his brethren; and the lad was with the sons of Bilhah, and with the sons of Zilpah, his father's wives: and Joseph brought unto his father their evil report.}}<ref name=":39" /> {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| Now Israel loved Joseph more than all his children, because he was the son of his old age: and he made him a coat of many colours.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And when his brethren saw that their father loved him more than all his brethren, they hated him, and could not speak peaceably unto him.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And Joseph dreamed a dream, and he told it his brethren: and they hated him yet the more.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And he said unto them, Hear, I pray you, this dream which I have dreamed:}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| For, behold, we were binding sheaves in the field, and, lo, my sheaf arose, and also stood upright; and, behold, your sheaves stood round about, and made obeisance to my sheaf.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And his brethren said to him, Shalt thou indeed reign over us? or shalt thou indeed have dominion over us? And they hated him yet the more for his dreams, and for his words.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And he dreamed yet another dream, and told it his brethren, and said, Behold, I have dreamed a dream more; and, behold, the sun and the moon and the eleven stars made obeisance to me.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he told it to his father, and to his brethren: and his father rebuked him, and said unto him, What is this dream that thou hast dreamed? Shall I and thy mother and thy brethren indeed come to bow down ourselves to thee to the earth?}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And his brethren envied him;}}<ref name=":39" /> {{font|color=#008888|but his father observed the saying.}}<ref name=":39" /> {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And his brethren went to feed their father's flock in Shechem.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And Israel said unto Joseph, Do not thy brethren feed the flock in Shechem? come, and I will send thee unto them. And he said to him, Here am I.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And he said to him, Go, I pray thee, see whether it be well with thy brethren, and well with the flocks; and bring me word again. So he sent him out of the vale of}}<ref name=":40">{{Cite web|url=https://twitter.com/joelbaden/status/1315253669072457729|title=Genesis 37:12-17|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|Hebron,}}<ref name=":40" /> {{font|color=#008888|and he came to Shechem.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| And a certain man found him, and, behold, he was wandering in the field: and the man asked him, saying, What seekest thou?}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| And he said, I seek my brethren: tell me, I pray thee, where they feed their flocks.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And the man said, They are departed hence; for I heard them say, Let us go to Dothan. And Joseph went after his brethren, and found them in Dothan.}}<ref name=":40" /> {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And when they saw him afar off, even before he came near unto them, they conspired against him to slay him.}}<ref name=":41">{{Cite web|url=https://twitter.com/joelbaden/status/1315634632352727041|title=Genesis 37:18-36|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And they said one to another, Behold, this dreamer cometh.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| Come now therefore, and let us slay him, and cast him into some pit, and we will say, Some evil beast hath devoured him: and we shall see what will become of his dreams.}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And Reuben heard it, and he delivered him out of their hands; and said, Let us not kill him.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| And Reuben said unto them, Shed no blood, but cast him into this pit that is in the wilderness, and lay no hand upon him; that he might rid him out of their hands, to deliver him to his father again.}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And it came to pass, when Joseph was come unto his brethren, that they stript Joseph out of his coat, his coat of many colours that was on him;}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And they took him, and cast him into a pit: and the pit was empty, there was no water in it.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| And they sat down to eat bread:}}<ref name=":41" /> {{font|color=#000088| and they lifted up their eyes and looked, and, behold, a company of Ishmeelites came from Gilead with their camels bearing spicery and balm and myrrh, going to carry it down to Egypt.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And Judah said unto his brethren, What profit is it if we slay our brother, and conceal his blood?}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| Come, and let us sell him to the Ishmeelites, and let not our hand be upon him; for he is our brother and our flesh. And his brethren were content.}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| Then there passed by Midianites merchantmen; and they drew and lifted up Joseph out of the pit,}}<ref name=":41" /> {{font|color=#000088| and sold Joseph to the Ishmeelites for twenty pieces of silver: and they brought Joseph into Egypt.}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|29}}{{font|color=#008888| And Reuben returned unto the pit; and, behold, Joseph was not in the pit; and he rent his clothes.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#008888| And he returned unto his brethren, and said, The child is not; and I, whither shall I go?}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And they took Joseph's coat, and killed a kid of the goats, and dipped the coat in the blood;}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And they sent the coat of many colours, and they brought it to their father; and said, This have we found: know now whether it be thy son's coat or no.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And he knew it, and said, It is my son's coat; an evil beast hath devoured him; Joseph is without doubt rent in pieces.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And Jacob rent his clothes, and put sackcloth upon his loins, and mourned for his son many days.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And all his sons and all his daughters rose up to comfort him; but he refused to be comforted; and he said, For I will go down into the grave unto my son mourning. Thus his father wept for him.}}<ref name=":41" /> {{font|size=smaller|color=#0000FF|36}}{{font|color=#008888| And the Midianites sold him into Egypt unto Potiphar, an officer of Pharaoh's, and captain of the guard.}}<ref name=":41" /> ==Chapter 38== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And it came to pass at that time, that Judah went down from his brethren, and turned in to a certain Adullamite, whose name was Hirah.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Judah saw there a daughter of a certain Canaanite, whose name was Shuah; and he took her, and went in unto her.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And she conceived, and bare a son; and he called his name Er.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And she conceived again, and bare a son; and she called his name Onan.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And she yet again conceived, and bare a son; and called his name Shelah: and he was at Chezib, when she bare him.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Judah took a wife for Er his firstborn, whose name was Tamar.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Er, Judah's firstborn, was wicked in the sight of the LORD; and the LORD slew him.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Judah said unto Onan, Go in unto thy brother's wife, and marry her, and raise up seed to thy brother.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Onan knew that the seed should not be his; and it came to pass, when he went in unto his brother's wife, that he spilled it on the ground, lest that he should give seed to his brother.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And the thing which he did displeased the LORD: wherefore he slew him also.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Then said Judah to Tamar his daughter in law, Remain a widow at thy father's house, till Shelah my son be grown: for he said, Lest peradventure he die also, as his brethren did. And Tamar went and dwelt in her father's house.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1315978404940972032|title=Genesis 38:1-11|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And in process of time the daughter of Shuah Judah's wife died; and Judah was comforted, and went up unto his sheepshearers to Timnath, he and his friend Hirah the Adullamite.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And it was told Tamar, saying, Behold thy father in law goeth up to Timnath to shear his sheep.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And she put her widow's garments off from her, and covered her with a vail, and wrapped herself, and sat in an open place, which is by the way to Timnath; for she saw that Shelah was grown, and she was not given unto him to wife.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| When Judah saw her, he thought her to be an harlot; because she had covered her face.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And he turned unto her by the way, and said, Go to, I pray thee, let me come in unto thee; (for he knew not that she was his daughter in law.) And she said, What wilt thou give me, that thou mayest come in unto me?}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And he said, I will send thee a kid from the flock. And she said, Wilt thou give me a pledge, till thou send it?}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And he said, What pledge shall I give thee? And she said, Thy signet, and thy bracelets, and thy staff that is in thine hand. And he gave it her, and came in unto her, and she conceived by him.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And she arose, and went away, and laid by her vail from her, and put on the garments of her widowhood.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Judah sent the kid by the hand of his friend the Adullamite, to receive his pledge from the woman's hand: but he found her not.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Then he asked the men of that place, saying, Where is the harlot, that was openly by the way side? And they said, There was no harlot in this place.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And he returned to Judah, and said, I cannot find her; and also the men of the place said, that there was no harlot in this place.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And Judah said, Let her take it to her, lest we be shamed: behold, I sent this kid, and thou hast not found her.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And it came to pass about three months after, that it was told Judah, saying, Tamar thy daughter in law hath played the harlot; and also, behold, she is with child by whoredom. And Judah said, Bring her forth, and let her be burnt.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| When she was brought forth, she sent to her father in law, saying, By the man, whose these are, am I with child: and she said, Discern, I pray thee, whose are these, the signet, and bracelets, and staff.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And Judah acknowledged them, and said, She hath been more righteous than I; because that I gave her not to Shelah my son. And he knew her again no more.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And it came to pass in the time of her travail, that, behold, twins were in her womb.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And it came to pass, when she travailed, that the one put out his hand: and the midwife took and bound upon his hand a scarlet thread, saying, This came out first.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And it came to pass, as he drew back his hand, that, behold, his brother came out: and she said, How hast thou broken forth? this breach be upon thee: therefore his name was called Pharez.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And afterward came out his brother, that had the scarlet thread upon his hand: and his name was called Zarah.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1316385230979293191|title=Genesis 38:12-30|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> ==Chapter 39== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Joseph was brought down to Egypt; and}}<ref name=":42">{{Cite web|url=https://twitter.com/joelbaden/status/1316728844980715523|title=Genesis 39:1|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|Potiphar, an officer of Pharaoh, captain of the guard,}}<ref name=":42" /> {{font|color=#000088|an Egyptian, bought him of the hands of the Ishmeelites, which had brought him down thither.}}<ref name=":42" /> {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And the LORD was with Joseph, and he was a prosperous man; and he was in the house of his master the Egyptian.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And his master saw that the LORD was with him, and that the LORD made all that he did to prosper in his hand.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And Joseph found grace in his sight, and he served him: and he made him overseer over his house, and all that he had he put into his hand.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And it came to pass from the time that he had made him overseer in his house, and over all that he had, that the LORD blessed the Egyptian's house for Joseph's sake; and the blessing of the LORD was upon all that he had in the house, and in the field.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And he left all that he had in Joseph's hand; and he knew not ought he had, save the bread which he did eat. And Joseph was a goodly person, and well favoured.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And it came to pass after these things, that his master's wife cast her eyes upon Joseph; and she said, Lie with me.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| But he refused, and said unto his master's wife, Behold, my master wotteth not what is with me in the house, and he hath committed all that he hath to my hand;}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| There is none greater in this house than I; neither hath he kept back any thing from me but thee, because thou art his wife: how then can I do this great wickedness, and sin against God?}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And it came to pass, as she spake to Joseph day by day, that he hearkened not unto her, to lie by her, or to be with her.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And it came to pass about this time, that Joseph went into the house to do his business; and there was none of the men of the house there within.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And she caught him by his garment, saying, Lie with me: and he left his garment in her hand, and fled, and got him out.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And it came to pass, when she saw that he had left his garment in her hand, and was fled forth,}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| That she called unto the men of her house, and spake unto them, saying, See, he hath brought in an Hebrew unto us to mock us; he came in unto me to lie with me, and I cried with a loud voice:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And it came to pass, when he heard that I lifted up my voice and cried, that he left his garment with me, and fled, and got him out.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And she laid up his garment by her, until his lord came home.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And she spake unto him according to these words, saying, The Hebrew servant, which thou hast brought unto us, came in unto me to mock me:}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And it came to pass, as I lifted up my voice and cried, that he left his garment with me, and fled out.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And it came to pass, when his master heard the words of his wife, which she spake unto him, saying, After this manner did thy servant to me; that his wrath was kindled.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Joseph's master took him, and put him into the prison, a place where the king's prisoners were bound: and he was there in the prison.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| But the LORD was with Joseph, and shewed him mercy, and gave him favour in the sight of the keeper of the prison.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And the keeper of the prison committed to Joseph's hand all the prisoners that were in the prison; and whatsoever they did there, he was the doer of it.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| The keeper of the prison looked not to any thing that was under his hand; because the LORD was with him, and that which he did, the LORD made it to prosper.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1317074681875341313|title=Genesis 39:2-23|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> ==Chapter 40== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And it came to pass after these things, that the butler of the king of Egypt and his baker had offended their lord the king of Egypt.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And Pharaoh was wroth against two of his officers, against the chief of the butlers, and against the chief of the bakers.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And he put them in ward in the house of the captain of the guard,}}<ref name=":43">{{Cite web|url=https://twitter.com/joelbaden/status/1317420602253299713|title=Genesis 40:1-4|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|into the prison, the place where Joseph was bound.}}<ref name=":43" /> {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| And the captain of the guard charged Joseph with them, and he served them: and they continued a season in ward.}}<ref name=":43" /> {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And they dreamed a dream both of them, each man his dream in one night, each man according to the interpretation of his dream, the butler and the baker of the king of Egypt, }}<ref name=":44">{{Cite web|url=https://twitter.com/joelbaden/status/1317807450909822977|title=Genesis 40:5-23|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|which were bound in the prison.}}<ref name=":43" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And Joseph came in unto them in the morning, and looked upon them, and, behold, they were sad.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| And he asked Pharaoh's officers that were with him in the ward of his lord's house, saying, Wherefore look ye so sadly to day?}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And they said unto him, We have dreamed a dream, and there is no interpreter of it. And Joseph said unto them, Do not interpretations belong to God? tell me them, I pray you.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And the chief butler told his dream to Joseph, and said to him, In my dream, behold, a vine was before me;}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| And in the vine were three branches: and it was as though it budded, and her blossoms shot forth; and the clusters thereof brought forth ripe grapes:}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And Pharaoh's cup was in my hand: and I took the grapes, and pressed them into Pharaoh's cup, and I gave the cup into Pharaoh's hand.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And Joseph said unto him, This is the interpretation of it: The three branches are three days:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| Yet within three days shall Pharaoh lift up thine head, and restore thee unto thy place: and thou shalt deliver Pharaoh's cup into his hand, after the former manner when thou wast his butler.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| But think on me when it shall be well with thee, and shew kindness, I pray thee, unto me, and make mention of me unto Pharaoh, and bring me out of this house:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| For indeed I was stolen away out of the land of the Hebrews:}}<ref name=":44" /> {{font|color=#880000|and here also have I done nothing that they should put me into the dungeon.}}<ref name=":44" /> {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| When the chief baker saw that the interpretation was good, he said unto Joseph, I also was in my dream, and, behold, I had three white baskets on my head:}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And in the uppermost basket there was of all manner of bakemeats for Pharaoh; and the birds did eat them out of the basket upon my head.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Joseph answered and said, This is the interpretation thereof: The three baskets are three days:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| Yet within three days shall Pharaoh lift up thy head from off thee, and shall hang thee on a tree; and the birds shall eat thy flesh from off thee.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And it came to pass the third day, which was Pharaoh's birthday, that he made a feast unto all his servants: and he lifted up the head of the chief butler and of the chief baker among his servants.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And he restored the chief butler unto his butlership again; and he gave the cup into Pharaoh's hand:}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| But he hanged the chief baker: as Joseph had interpreted to them.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| Yet did not the chief butler remember Joseph, but forgat him.}}<ref name=":44" /> ==Chapter 41== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And it came to pass at the end of two full years, that Pharaoh dreamed: and, behold, he stood by the river.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And, behold, there came up out of the river seven well favoured kine and fatfleshed; and they fed in a meadow.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And, behold, seven other kine came up after them out of the river, ill favoured and leanfleshed; and stood by the other kine upon the brink of the river.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| And the ill favoured and leanfleshed kine did eat up the seven well favoured and fat kine. So Pharaoh awoke.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And he slept and dreamed the second time: and, behold, seven ears of corn came up upon one stalk, rank and good.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#008888| And, behold, seven thin ears and blasted with the east wind sprung up after them.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#008888| And the seven thin ears devoured the seven rank and full ears. And Pharaoh awoke, and, behold, it was a dream.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And it came to pass in the morning that his spirit was troubled; and he sent and called for all the magicians of Egypt, and all the wise men thereof: and Pharaoh told them his dream; but there was none that could interpret them unto Pharaoh.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| Then spake the chief butler unto Pharaoh, saying, I do remember my faults this day:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| Pharaoh was wroth with his servants, and put me in ward in the captain of the guard's house, both me and the chief baker:}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And we dreamed a dream in one night, I and he; we dreamed each man according to the interpretation of his dream.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And there was there with us a young man, an Hebrew, servant to the captain of the guard; and we told him, and he interpreted to us our dreams; to each man according to his dream he did interpret.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And it came to pass, as he interpreted to us, so it was; me he restored unto mine office, and him he hanged.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1318157819770769408|title=Genesis 41:1-13|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| Then Pharaoh sent and called Joseph, and they brought him hastily out of the dungeon: and he shaved himself, and changed his raiment, and came in unto Pharaoh.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| And Pharaoh said unto Joseph, I have dreamed a dream, and there is none that can interpret it: and I have heard say of thee, that thou canst understand a dream to interpret it.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| And Joseph answered Pharaoh, saying, It is not in me: God shall give Pharaoh an answer of peace.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And Pharaoh said unto Joseph, In my dream, behold, I stood upon the bank of the river:}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And, behold, there came up out of the river seven kine, fatfleshed and well favoured; and they fed in a meadow:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And, behold, seven other kine came up after them, poor and very ill favoured and leanfleshed, such as I never saw in all the land of Egypt for badness:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#008888| And the lean and the ill favoured kine did eat up the first seven fat kine:}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And when they had eaten them up, it could not be known that they had eaten them; but they were still ill favoured, as at the beginning. So I awoke.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| And I saw in my dream, and, behold, seven ears came up in one stalk, full and good:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| And, behold, seven ears, withered, thin, and blasted with the east wind, sprung up after them:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And the thin ears devoured the seven good ears: and I told this unto the magicians; but there was none that could declare it to me.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| And Joseph said unto Pharaoh, The dream of Pharaoh is one: God hath shewed Pharaoh what he is about to do.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#008888| The seven good kine are seven years; and the seven good ears are seven years: the dream is one.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#008888| And the seven thin and ill favoured kine that came up after them are seven years; and the seven empty ears blasted with the east wind shall be seven years of famine.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#008888| This is the thing which I have spoken unto Pharaoh: What God is about to do he sheweth unto Pharaoh.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#008888| Behold, there come seven years of great plenty throughout all the land of Egypt:}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#008888| And there shall arise after them seven years of famine; and all the plenty shall be forgotten in the land of Egypt; and the famine shall consume the land;}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#008888| And the plenty shall not be known in the land by reason of that famine following; for it shall be very grievous.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#008888| And for that the dream was doubled unto Pharaoh twice; it is because the thing is established by God, and God will shortly bring it to pass.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#008888| Now therefore let Pharaoh look out a man discreet and wise, and set him over the land of Egypt.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#008888| Let Pharaoh do this, and let him appoint officers over the land, and take up the fifth part of the land of Egypt in the seven plenteous years.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#008888| And let them gather all the food of those good years that come, and lay up corn under the hand of Pharaoh, and let them keep food in the cities.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#008888| And that food shall be for store to the land against the seven years of famine, which shall be in the land of Egypt; that the land perish not through the famine.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1318506616908546053|title=Genesis 41:14-36|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|37}}{{font|color=#008888| And the thing was good in the eyes of Pharaoh, and in the eyes of all his servants.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#008888| And Pharaoh said unto his servants, Can we find such a one as this is, a man in whom the Spirit of God is?}} {{font|size=smaller|color=#0000FF|39}}{{font|color=#008888| And Pharaoh said unto Joseph, Forasmuch as God hath shewed thee all this, there is none so discreet and wise as thou art:}} {{font|size=smaller|color=#0000FF|40}}{{font|color=#008888| Thou shalt be over my house, and according unto thy word shall all my people be ruled: only in the throne will I be greater than thou.}}<ref name=":45">{{Cite web|url=https://twitter.com/joelbaden/status/1318893501753511937|title=Genesis 41:37-46|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|size=smaller|color=#0000FF|41}}{{font|color=#000088| And Pharaoh said unto Joseph, See, I have set thee over all the land of Egypt.}} {{font|size=smaller|color=#0000FF|42}}{{font|color=#000088| And Pharaoh took off his ring from his hand, and put it upon Joseph's hand, and arrayed him in vestures of fine linen, and put a gold chain about his neck;}} {{font|size=smaller|color=#0000FF|43}}{{font|color=#000088| And he made him to ride in the second chariot which he had; and they cried before him, Bow the knee: and he made him ruler over all the land of Egypt.}}<ref name=":45" /> {{font|size=smaller|color=#0000FF|44}}{{font|color=#888800| And Pharaoh said unto Joseph, I am Pharaoh, and without thee shall no man lift up his hand or foot in all the land of Egypt.}} {{font|size=smaller|color=#0000FF|45}}{{font|color=#888800| And Pharaoh called Joseph's name Zaphnathpaaneah; and he gave him to wife Asenath the daughter of Potipherah priest of On.}}<ref name=":45" /> {{font|color=#000088| And Joseph went out over all the land of Egypt.}}<ref name=":45" /> {{font|size=smaller|color=#0000FF|46}}{{font|color=#888800| And Joseph was thirty years old when he stood before Pharaoh king of Egypt.}}<ref name=":45" /> {{font|color=#008888| And Joseph went out from the presence of Pharaoh, and went throughout all the land of Egypt.}}<ref name=":45" /> {{font|size=smaller|color=#0000FF|47}}{{font|color=#008888| And in the seven plenteous years the earth brought forth by handfuls.}} {{font|size=smaller|color=#0000FF|48}}{{font|color=#008888| And he gathered up all the food of the seven years, which were in the land of Egypt, and laid up the food in the cities: the food of the field, which was round about every city, laid he up in the same.}} {{font|size=smaller|color=#0000FF|49}}{{font|color=#008888| And Joseph gathered corn as the sand of the sea, very much, until he left numbering; for it was without number.}} {{font|size=smaller|color=#0000FF|50}}{{font|color=#008888| And unto Joseph were born two sons before the years of famine came,}}<ref name=":46">{{Cite web|url=https://twitter.com/joelbaden/status/1319250773436489728|title=Genesis 41:47-52|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-15}}</ref> {{font|color=#880000|which Asenath the daughter of Potipherah priest of On bare unto him.}}<ref name=":46" /> {{font|size=smaller|color=#0000FF|51}}{{font|color=#008888| And Joseph called the name of the firstborn Manasseh: For God, said he, hath made me forget all my toil, and all my father's house.}} {{font|size=smaller|color=#0000FF|52}}{{font|color=#008888| And the name of the second called he Ephraim: For God hath caused me to be fruitful in the land of my affliction.}}<ref name=":46" /> {{font|size=smaller|color=#0000FF|53}}{{font|color=#008888| And the seven years of plenteousness, that was in the land of Egypt, were ended.}} {{font|size=smaller|color=#0000FF|54}}{{font|color=#008888| And the seven years of dearth began to come, according as Joseph had said:}}<ref name=":47">{{Cite web|url=https://twitter.com/joelbaden/status/1319611144705875970|title=Genesis 41:53-57|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-17}}</ref> {{font|color=#000088|and the dearth was in all lands; but in all the land of Egypt there was bread.}}<ref name=":47" /> {{font|size=smaller|color=#0000FF|55}}{{font|color=#008888| And when all the land of Egypt was famished, the people cried to Pharaoh for bread: and Pharaoh said unto all the Egyptians, Go unto Joseph; what he saith to you, do.}}<ref name=":47" /> {{font|size=smaller|color=#0000FF|56}}{{font|color=#000088| And the famine was over all the face of the earth:}}<ref name=":47" /> {{font|color=#008888|and Joseph opened all the storehouses, and sold unto the Egyptians; and the famine waxed sore in the land of Egypt.}}<ref name=":47" /> {{font|size=smaller|color=#0000FF|57}}{{font|color=#000088| And all countries came into Egypt}}<ref name=":47" /> {{font|color=#880000|to Joseph}}<ref name=":47" /> {{font|color=#000088|for to buy corn; because that the famine was so sore in all lands.}}<ref name=":47" /> ==Chapter 42== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Now when Jacob saw that there was corn in Egypt, Jacob said unto his sons, Why do ye look one upon another?}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he said, Behold, I have heard that there is corn in Egypt: get you down thither, and buy for us from thence; that we may live, and not die.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Joseph's ten brethren went down to buy corn in Egypt.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| But Benjamin, Joseph's brother, Jacob sent not with his brethren; for he said, Lest peradventure mischief befall him.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| And the sons of Israel came to buy corn among those that came: for the famine was in the land of Canaan.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Joseph was the governor over the land, and he it was that sold to all the people of the land: and Joseph's brethren came, and bowed down themselves before him with their faces to the earth.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Joseph saw his brethren, and he knew them, but made himself strange unto them, and spake roughly unto them; and he said unto them, Whence come ye? And they said, From the land of Canaan to buy food.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Joseph knew his brethren, but they knew not him.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And Joseph remembered the dreams which he dreamed of them, and said unto them, Ye are spies; to see the nakedness of the land ye are come.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And they said unto him, Nay, my lord, but to buy food are thy servants come.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| We are all one man's sons; we are true men, thy servants are no spies.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he said unto them, Nay, but to see the nakedness of the land ye are come.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And they said, Thy servants are twelve brethren, the sons of one man in the land of Canaan; and, behold, the youngest is this day with our father, and one is not.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Joseph said unto them, That is it that I spake unto you, saying, Ye are spies:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| Hereby ye shall be proved: By the life of Pharaoh ye shall not go forth hence, except your youngest brother come hither.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| Send one of you, and let him fetch your brother, and ye shall be kept in prison, that your words may be proved, whether there be any truth in you: or else by the life of Pharaoh surely ye are spies.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And he put them all together into ward three days.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And Joseph said unto them the third day, This do, and live; for I fear God:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| If ye be true men, let one of your brethren be bound in the house of your prison: go ye, carry corn for the famine of your houses:}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| But bring your youngest brother unto me; so shall your words be verified, and ye shall not die. And they did so.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And they said one to another, We are verily guilty concerning our brother, in that we saw the anguish of his soul, when he besought us, and we would not hear; therefore is this distress come upon us.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Reuben answered them, saying, Spake I not unto you, saying, Do not sin against the child; and ye would not hear? therefore, behold, also his blood is required.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And they knew not that Joseph understood them; for he spake unto them by an interpreter.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And he turned himself about from them, and wept; and returned to them again, and communed with them, and took from them Simeon, and bound him before their eyes.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| Then Joseph commanded to fill their sacks with corn, and to restore every man's money into his sack, and to give them provision for the way: and thus did he unto them.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And they laded their asses with the corn, and departed thence.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And as one of them opened his sack to give his ass provender in the inn, he espied his money; for, behold, it was in his sack's mouth.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And he said unto his brethren, My money is restored; and, lo, it is even in my sack: and their heart failed them, and they were afraid, saying one to another, What is this that God hath done unto us?}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And they came unto Jacob their father unto the land of Canaan, and told him all that befell unto them; saying,}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| The man, who is the lord of the land, spake roughly to us, and took us for spies of the country.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And we said unto him, We are true men; we are no spies:}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| We be twelve brethren, sons of our father; one is not, and the youngest is this day with our father in the land of Canaan.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And the man, the lord of the country, said unto us, Hereby shall I know that ye are true men; leave one of your brethren here with me, and take food for the famine of your households, and be gone:}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And bring your youngest brother unto me: then shall I know that ye are no spies, but that ye are true men: so will I deliver you your brother, and ye shall traffick in the land.}} {{font|size=smaller|color=#0000FF|35}}{{font|color=#000088| And it came to pass as they emptied their sacks, that, behold, every man's bundle of money was in his sack: and when both they and their father saw the bundles of money, they were afraid.}} {{font|size=smaller|color=#0000FF|36}}{{font|color=#000088| And Jacob their father said unto them, Me have ye bereaved of my children: Joseph is not, and Simeon is not, and ye will take Benjamin away: all these things are against me.}} {{font|size=smaller|color=#0000FF|37}}{{font|color=#000088| And Reuben spake unto his father, saying, Slay my two sons, if I bring him not to thee: deliver him into my hand, and I will bring him to thee again.}} {{font|size=smaller|color=#0000FF|38}}{{font|color=#000088| And he said, My son shall not go down with you; for his brother is dead, and he is left alone: if mischief befall him by the way in the which ye go, then shall ye bring down my gray hairs with sorrow to the grave.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1319950576059371520|title=Genesis 42|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> ==Chapter 43== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And the famine was sore in the land.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And it came to pass, when they had eaten up the corn which they had brought out of Egypt, their father said unto them, Go again, buy us a little food.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Judah spake unto him, saying, The man did solemnly protest unto us, saying, Ye shall not see my face, except your brother be with you.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| If thou wilt send our brother with us, we will go down and buy thee food:}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| But if thou wilt not send him, we will not go down: for the man said unto us, Ye shall not see my face, except your brother be with you.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Israel said, Wherefore dealt ye so ill with me, as to tell the man whether ye had yet a brother?}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And they said, The man asked us straitly of our state, and of our kindred, saying, Is your father yet alive? have ye another brother? and we told him according to the tenor of these words: could we certainly know that he would say, Bring your brother down?}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And Judah said unto Israel his father, Send the lad with me, and we will arise and go; that we may live, and not die, both we, and thou, and also our little ones.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| I will be surety for him; of my hand shalt thou require him: if I bring him not unto thee, and set him before thee, then let me bear the blame for ever:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| For except we had lingered, surely now we had returned this second time.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And their father Israel said unto them, If it must be so now, do this; take of the best fruits in the land in your vessels, and carry down the man a present, a little balm, and a little honey, spices, and myrrh, nuts, and almonds:}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And take double money in your hand; and the money that was brought again in the mouth of your sacks, carry it again in your hand; peradventure it was an oversight:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| Take also your brother, and arise, go again unto the man:}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And God Almighty give you mercy before the man, that he may send away your other brother, and Benjamin. If I be bereaved of my children, I am bereaved.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And the men took that present, and they took double money in their hand, and Benjamin; and rose up, and went down to Egypt, and stood before Joseph.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And when Joseph saw Benjamin with them, he said to the ruler of his house, Bring these men home, and slay, and make ready; for these men shall dine with me at noon.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And the man did as Joseph bade; and the man brought the men into Joseph's house.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And the men were afraid, because they were brought into Joseph's house; and they said, Because of the money that was returned in our sacks at the first time are we brought in; that he may seek occasion against us, and fall upon us, and take us for bondmen, and our asses.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And they came near to the of Joseph's house, and they communed with him at the door of the house,}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And said, O sir, we came indeed down at the first time to buy food:}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And it came to pass, when we came to the inn, that we opened our sacks, and, behold, every man's money was in the mouth of his sack, our money in full weight: and we have brought it again in our hand.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And other money have we brought down in our hands to buy food: we cannot tell who put our money in our sacks.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And he said, Peace be to you, fear not: your God, and the God of your father, hath given you treasure in your sacks: I had your money. And he brought Simeon out unto them.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And the man brought the men into Joseph's house, and gave them water, and they washed their feet; and he gave their asses provender.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And they made ready the present against Joseph came at noon: for they heard that they should eat bread there.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And when Joseph came home, they brought him the present which was in their hand into the house, and bowed themselves to him to the earth.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And he asked them of their welfare, and said, Is your father well, the old man of whom ye spake? Is he yet alive?}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And they answered, Thy servant our father is in good health, he is yet alive. And they bowed down their heads, and made obeisance.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And he lifted up his eyes, and saw his brother Benjamin, his mother's son, and said, Is this your younger brother, of whom ye spake unto me? And he said, God be gracious unto thee, my son.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And Joseph made haste; for his bowels did yearn upon his brother: and he sought where to weep; and he entered into his chamber, and wept there.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he washed his face, and went out, and refrained himself, and said, Set on bread.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And they set on for him by himself, and for them by themselves, and for the Egyptians, which did eat with him, by themselves: because the Egyptians might not eat bread with the Hebrews; for that is an abomination unto the Egyptians.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And they sat before him, the firstborn according to his birthright, and the youngest according to his youth: and the men marvelled one at another.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| And he took and sent messes unto them from before him: but Benjamin's mess was five times so much as any of theirs. And they drank, and were merry with him.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1320324229246689280|title=Genesis 43|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> ==Chapter 44== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And he commanded the steward of his house, saying, Fill the men's sacks with food, as much as they can carry, and put every man's money in his sack's mouth.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And put my cup, the silver cup, in the sack's mouth of the youngest, and his corn money. And he did according to the word that Joseph had spoken.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| As soon as the morning was light, the men were sent away, they and their asses.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And when they were gone out of the city, and not yet far off, Joseph said unto his steward, Up, follow after the men; and when thou dost overtake them, say unto them, Wherefore have ye rewarded evil for good?}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| Is not this it in which my lord drinketh, and whereby indeed he divineth? ye have done evil in so doing.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And he overtook them, and he spake unto them these same words.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And they said unto him, Wherefore saith my lord these words? God forbid that thy servants should do according to this thing:}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| Behold, the money, which we found in our sacks' mouths, we brought again unto thee out of the land of Canaan: how then should we steal out of thy lord's house silver or gold?}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| With whomsoever of thy servants it be found, both let him die, and we also will be my lord's bondmen.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And he said, Now also let it be according unto your words: he with whom it is found shall be my servant; and ye shall be blameless.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Then they speedily took down every man his sack to the ground, and opened every man his sack.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And he searched, and began at the eldest, and left at the youngest: and the cup was found in Benjamin's sack.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| Then they rent their clothes, and laded every man his ass, and returned to the city.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Judah and his brethren came to Joseph's house; for he was yet there: and they fell before him on the ground.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And Joseph said unto them, What deed is this that ye have done? wot ye not that such a man as I can certainly divine?}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Judah said, What shall we say unto my lord? what shall we speak? or how shall we clear ourselves? God hath found out the iniquity of thy servants: behold, we are my lord's servants, both we, and he also with whom the cup is found.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And he said, God forbid that I should do so: but the man in whose hand the cup is found, he shall be my servant; and as for you, get you up in peace unto your father.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| Then Judah came near unto him, and said, Oh my lord, let thy servant, I pray thee, speak a word in my lord's ears, and let not thine anger burn against thy servant: for thou art even as Pharaoh.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| My lord asked his servants, saying, Have ye a father, or a brother?}}v {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And we said unto my lord, We have a father, an old man, and a child of his old age, a little one; and his brother is dead, and he alone is left of his mother, and his father loveth him.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And thou saidst unto thy servants, Bring him down unto me, that I may set mine eyes upon him.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And we said unto my lord, The lad cannot leave his father: for if he should leave his father, his father would die.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And thou saidst unto thy servants, Except your youngest brother come down with you, ye shall see my face no more.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And it came to pass when we came up unto thy servant my father, we told him the words of my lord.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And our father said, Go again, and buy us a little food.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And we said, We cannot go down: if our youngest brother be with us, then will we go down: for we may not see the man's face, except our youngest brother be with us.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And thy servant my father said unto us, Ye know that my wife bare me two sons:}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And the one went out from me, and I said, Surely he is torn in pieces; and I saw him not since:}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And if ye take this also from me, and mischief befall him, ye shall bring down my gray hairs with sorrow to the grave.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| Now therefore when I come to thy servant my father, and the lad be not with us; seeing that his life is bound up in the lad's life;}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| It shall come to pass, when he seeth that the lad is not with us, that he will die: and thy servants shall bring down the gray hairs of thy servant our father with sorrow to the grave.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| For thy servant became surety for the lad unto my father, saying, If I bring him not unto thee, then I shall bear the blame to my father for ever.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| Now therefore, I pray thee, let thy servant abide instead of the lad a bondman to my lord; and let the lad go up with his brethren.}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| For how shall I go up to my father, and the lad be not with me? lest peradventure I see the evil that shall come on my father.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1320680539444203520|title=Genesis 44|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> ==Chapter 45== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Then Joseph could not refrain himself before all them that stood by him; and he cried, Cause every man to go out from me. And there stood no man with him, while Joseph made himself known unto his brethren.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he wept aloud: and the Egyptians and the house of Pharaoh heard.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Joseph said unto his brethren, I am Joseph; doth my father yet live? And his brethren could not answer him; for they were troubled at his presence.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And Joseph said unto his brethren, Come near to me, I pray you. And they came near. And he said, I am Joseph your brother, whom ye sold into Egypt.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| Now therefore be not grieved, nor angry with yourselves, that ye sold me hither: for God did send me before you to preserve life.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| For these two years hath the famine been in the land: and yet there are five years, in the which there shall neither be earing nor harvest.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And God sent me before you to preserve you a posterity in the earth, and to save your lives by a great deliverance.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| So now it was not you that sent me hither, but God: and he hath made me a father to Pharaoh, and lord of all his house, and a ruler throughout all the land of Egypt.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| Haste ye, and go up to my father, and say unto him, Thus saith thy son Joseph, God hath made me lord of all Egypt: come down unto me, tarry not:}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And thou shalt dwell in the land of Goshen, and thou shalt be near unto me, thou, and thy children, and thy children's children, and thy flocks, and thy herds, and all that thou hast:}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And there will I nourish thee; for yet there are five years of famine; lest thou, and thy household, and all that thou hast, come to poverty.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And, behold, your eyes see, and the eyes of my brother Benjamin, that it is my mouth that speaketh unto you.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And ye shall tell my father of all my glory in Egypt, and of all that ye have seen; and ye shall haste and bring down my father hither.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And he fell upon his brother Benjamin's neck, and wept; and Benjamin wept upon his neck.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| Moreover he kissed all his brethren, and wept upon them: and after that his brethren talked with him.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And the fame thereof was heard in Pharaoh's house, saying, Joseph's brethren are come: and it pleased Pharaoh well, and his servants.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And Pharaoh said unto Joseph, Say unto thy brethren, This do ye; lade your beasts, and go, get you unto the land of Canaan;}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And take your father and your households, and come unto me: and I will give you the good of the land of Egypt, and ye shall eat the fat of the land.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| Now thou art commanded, this do ye; take you wagons out of the land of Egypt for your little ones, and for your wives, and bring your father, and come.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| Also regard not your stuff; for the good of all the land of Egypt is yours.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And the children of Israel did so: and Joseph gave them wagons, according to the commandment of Pharaoh, and gave them provision for the way.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| To all of them he gave each man changes of raiment; but to Benjamin he gave three hundred pieces of silver, and five changes of raiment.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| And to his father he sent after this manner; ten asses laden with the good things of Egypt, and ten she asses laden with corn and bread and meat for his father by the way.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| So he sent his brethren away, and they departed: and he said unto them, See that ye fall not out by the way.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And they went up out of Egypt, and came into the land of Canaan unto Jacob their father,}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And told him, saying, Joseph is yet alive, and he is governor over all the land of Egypt. And Jacob's heart fainted, for he believed them not.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| And they told him all the words of Joseph, which he had said unto them: and when he saw the wagons which Joseph had sent to carry him, the spirit of Jacob their father revived:}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And Israel said, It is enough; Joseph my son is yet alive: I will go and see him before I die.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1321065270711111681|title=Genesis 45|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> ==Chapter 46== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And Israel took his journey with all that he had, and came to Beersheba, and offered sacrifices unto the God of his father Isaac.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And God spake unto Israel in the visions of the night, and said, Jacob, Jacob. And he said, Here am I.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#008888| And he said, I am God, the God of thy father: fear not to go down into Egypt; for I will there make of thee a great nation:}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#008888| I will go down with thee into Egypt; and I will also surely bring thee up again: and Joseph shall put his hand upon thine eyes.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#008888| And Jacob rose up from Beersheba:}}<ref name=":48">{{Cite web|url=https://twitter.com/joelbaden/status/1321425430063304704|title=Genesis 46:1-7|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|color=#000088| and the sons of Israel carried Jacob their father, and their little ones, and their wives, in the wagons which Pharaoh had sent to carry him.}}<ref name=":48" /> {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And they took their cattle, and their goods, which they had gotten in the land of Canaan, and came into Egypt, Jacob, and all his seed with him:}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| His sons, and his sons' sons with him, his daughters, and his sons' daughters, and all his seed brought he with him into Egypt.}}<ref name=":48" /> {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And these are the names of the children of Israel, which came into Egypt, Jacob and his sons: Reuben, Jacob's firstborn.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And the sons of Reuben; Hanoch, and Phallu, and Hezron, and Carmi.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And the sons of Simeon; Jemuel, and Jamin, and Ohad, and Jachin, and Zohar, and Shaul the son of a Canaanitish woman.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And the sons of Levi; Gershon, Kohath, and Merari.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And the sons of Judah; Er, and Onan, and Shelah, and Pharez, and Zarah:}}<ref name=":49">{{Cite web|url=https://twitter.com/joelbaden/status/1321769774628642816|title=Genesis 46:8-27|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|color=#880000|but Er and Onan died in the land of Canaan.}}<ref name=":49" /> {{font|color=#888800|And the sons of Pharez were Hezron and Hamul.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| And the sons of Issachar; Tola, and Phuvah, and Job, and Shimron.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#888800| And the sons of Zebulun; Sered, and Elon, and Jahleel.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#888800| These be the sons of Leah, which she bare unto Jacob in Padanaram,}}<ref name=":49" /> {{font|color=#880000|with his daughter Dinah:}}<ref name=":49" /> {{font|color=#888800|all the souls of his sons and his daughters were thirty and three.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#888800| And the sons of Gad; Ziphion, and Haggi, Shuni, and Ezbon, Eri, and Arodi, and Areli.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#888800| And the sons of Asher; Jimnah, and Ishuah, and Isui, and Beriah, and Serah their sister: and the sons of Beriah; Heber, and Malchiel.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#888800| These are the sons of Zilpah, whom Laban gave to Leah his daughter, and these she bare unto Jacob, even sixteen souls.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#888800| The sons of Rachel Jacob's wife; Joseph, and Benjamin.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And unto Joseph in the land of Egypt were born Manasseh and Ephraim, which Asenath the daughter of Potipherah priest of On bare unto him.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#888800| And the sons of Benjamin were Belah, and Becher, and Ashbel, Gera, and Naaman, Ehi, and Rosh, Muppim, and Huppim, and Ard.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#888800| These are the sons of Rachel, which were born to Jacob: all the souls were fourteen.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#888800| And the sons of Dan; Hushim.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#888800| And the sons of Naphtali; Jahzeel, and Guni, and Jezer, and Shillem.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#888800| These are the sons of Bilhah, which Laban gave unto Rachel his daughter, and she bare these unto Jacob: all the souls were seven.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#888800| All the souls that came with Jacob into Egypt, which came out of his loins, besides Jacob's sons' wives, all the souls were threescore and six;}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| And the sons of Joseph, which were born him in Egypt, were two souls: all the souls of the house of Jacob, which came into Egypt, were threescore and ten.}}<ref name=":49" /> {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| And he sent Judah before him unto Joseph, to direct his face unto Goshen; and they came into the land of Goshen.}} {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And Joseph made ready his chariot, and went up to meet Israel his father, to Goshen, and presented himself unto him; and he fell on his neck, and wept on his neck a good while.}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| And Israel said unto Joseph, Now let me die, since I have seen thy face, because thou art yet alive.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And Joseph said unto his brethren, and unto his father's house, I will go up, and shew Pharaoh, and say unto him, My brethren, and my father's house, which were in the land of Canaan, are come unto me;}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#000088| And the men are shepherds, for their trade hath been to feed cattle; and they have brought their flocks, and their herds, and all that they have.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#000088| And it shall come to pass, when Pharaoh shall call you, and shall say, What is your occupation?}} {{font|size=smaller|color=#0000FF|34}}{{font|color=#000088| That ye shall say, Thy servants' trade hath been about cattle from our youth even until now, both we, and also our fathers: that ye may dwell in the land of Goshen; for every shepherd is an abomination unto the Egyptians.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1322107885913071616|title=Genesis 46:28-34|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> ==Chapter 47== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| Then Joseph came and told Pharaoh, and said, My father and my brethren, and their flocks, and their herds, and all that they have, are come out of the land of Canaan; and, behold, they are in the land of Goshen.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And he took some of his brethren, even five men, and presented them unto Pharaoh.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And Pharaoh said unto his brethren, What is your occupation? And they said unto Pharaoh, Thy servants are shepherds, both we, and also our fathers.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088|{{font|color=#000088| They said moreover unto Pharaoh, For to sojourn in the land are we come; for thy servants have no pasture for their flocks; for the famine is sore in the land of Canaan: now therefore, we pray thee, let thy servants dwell in the land of Goshen.}}}}<ref name=":50">{{Cite web|url=https://twitter.com/joelbaden/status/1322513069529047044|title=Genesis 47:1-12|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And Pharaoh spake unto Joseph, saying, Thy father and thy brethren are come unto thee:}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| The land of Egypt is before thee; in the best of the land make thy father and brethren to dwell;}}<ref name=":50" /> {{font|color=#000088|in the land of Goshen let them dwell: and if thou knowest any men of activity among them, then make them rulers over my cattle.}}<ref name=":50" /> {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And Joseph brought in Jacob his father, and set him before Pharaoh: and Jacob blessed Pharaoh.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#888800| And Pharaoh said unto Jacob, How old art thou?}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#888800| And Jacob said unto Pharaoh, The days of the years of my pilgrimage are an hundred and thirty years: few and evil have the days of the years of my life been, and have not attained unto the days of the years of the life of my fathers in the days of their pilgrimage.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#888800| And Jacob blessed Pharaoh, and went out from before Pharaoh.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#888800| And Joseph placed his father and his brethren, and gave them a possession in the land of Egypt, in the best of the land, in the land of Rameses, as Pharaoh had commanded.}}<ref name=":50" /> {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| And Joseph nourished his father, and his brethren, and all his father's household, with bread, according to their families.}}<ref name=":50" /> {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| And there was no bread in all the land; for the famine was very sore, so that the land of Egypt and all the land of Canaan fainted by reason of the famine.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Joseph gathered up all the money that was found in the land of Egypt, and in the land of Canaan, for the corn which they bought: and Joseph brought the money into Pharaoh's house.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And when money failed in the land of Egypt, and in the land of Canaan, all the Egyptians came unto Joseph, and said, Give us bread: for why should we die in thy presence? for the money faileth.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And Joseph said, Give your cattle; and I will give you for your cattle, if money fail.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| And they brought their cattle unto Joseph: and Joseph gave them bread in exchange for horses, and for the flocks, and for the cattle of the herds, and for the asses: and he fed them with bread for all their cattle for that year.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| When that year was ended, they came unto him the second year, and said unto him, We will not hide it from my lord, how that our money is spent; my lord also hath our herds of cattle; there is not ought left in the sight of my lord, but our bodies, and our lands:}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| Wherefore shall we die before thine eyes, both we and our land? buy us and our land for bread, and we and our land will be servants unto Pharaoh: and give us seed, that we may live, and not die, that the land be not desolate.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| And Joseph bought all the land of Egypt for Pharaoh; for the Egyptians sold every man his field, because the famine prevailed over them: so the land became Pharaoh's.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| And as for the people, he removed them to cities from one end of the borders of Egypt even to the other end thereof.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| Only the land of the priests bought he not; for the priests had a portion assigned them of Pharaoh, and did eat their portion which Pharaoh gave them: wherefore they sold not their lands.}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| Then Joseph said unto the people, Behold, I have bought you this day and your land for Pharaoh: lo, here is seed for you, and ye shall sow the land.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| And it shall come to pass in the increase, that ye shall give the fifth part unto Pharaoh, and four parts shall be your own, for seed of the field, and for your food, and for them of your households, and for food for your little ones.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| And they said, Thou hast saved our lives: let us find grace in the sight of my lord, and we will be Pharaoh's servants.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| And Joseph made it a law over the land of Egypt unto this day, that Pharaoh should have the fifth part; except the land of the priests only, which became not Pharaoh's.}}<ref>{{Cite web|url=https://twitter.com/joelbaden/status/1322874199514451971|title=Genesis 47:13-26|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|size=smaller|color=#0000FF|27}}{{font|color=#888800| And Israel dwelt in the land of Egypt,}}<ref name=":51">{{Cite web|url=https://twitter.com/joelbaden/status/1323218832908079104|title=Genesis 47:27-31|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|color=#880000|in the country of Goshen;}}<ref name=":51" /> {{font|color=#888800| and they had possessions therein, and grew, and multiplied exceedingly}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#888800| And Jacob lived in the land of Egypt seventeen years: so the whole age of Jacob was an hundred forty and seven years.}}<ref name=":51" /> {{font|size=smaller|color=#0000FF|29}}{{font|color=#000088| And the time drew nigh that Israel must die: and he called his son Joseph, and said unto him, If now I have found grace in thy sight, put, I pray thee, thy hand under my thigh, and deal kindly and truly with me; bury me not, I pray thee, in Egypt:}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#000088| But I will lie with my fathers, and thou shalt carry me out of Egypt, and bury me in their buryingplace. And he said, I will do as thou hast said.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#000088| And he said, Swear unto me. And he sware unto him. And Israel bowed himself upon the bed's head.}}<ref name=":51" /> ==Chapter 48== {{font|size=smaller|color=#0000FF|1}}{{font|color=#008888| And it came to pass after these things, that one told Joseph, Behold, thy father is sick: and he took with him his two sons, Manasseh and Ephraim.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#008888| And one told Jacob, and said, Behold, thy son Joseph cometh unto thee: and Israel strengthened himself, and sat upon the bed.}}<ref name=":52">{{Cite web|url=https://twitter.com/joelbaden/status/1324334990537183232|title=Genesis 48|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|size=smaller|color=#0000FF|3}}{{font|color=#888800| And Jacob said unto Joseph, God Almighty appeared unto me at Luz in the land of Canaan, and blessed me,}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#888800| And said unto me, Behold, I will make thee fruitful, and multiply thee, and I will make of thee a multitude of people; and will give this land to thy seed after thee for an everlasting possession.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#888800| And now thy two sons, Ephraim and Manasseh, which were born unto thee in the land of Egypt before I came unto thee into Egypt, are mine; as Reuben and Simeon, they shall be mine.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#888800| And thy issue, which thou begettest after them, shall be thine, and shall be called after the name of their brethren in their inheritance.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#888800| And as for me, when I came from Padan, Rachel died by me in the land of Canaan in the way, when yet there was but a little way to come unto Ephrath: and I buried her there in the way of Ephrath; the same is Bethlehem.}}<ref name=":52" /> {{font|size=smaller|color=#0000FF|8}}{{font|color=#008888| And Israel beheld Joseph's sons, and said, Who are these?}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#008888| And Joseph said unto his father, They are my sons, whom God hath given me in this place. And he said, Bring them, I pray thee, unto me, and I will bless them.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#008888| Now the eyes of Israel were dim for age, so that he could not see. And he brought them near unto him; and he kissed them, and embraced them.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#008888| And Israel said unto Joseph, I had not thought to see thy face: and, lo, God hath shewed me also thy seed.}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#008888| And Joseph brought them out from between his knees, and he bowed himself with his face to the earth.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#008888| And Joseph took them both, Ephraim in his right hand toward Israel's left hand, and Manasseh in his left hand toward Israel's right hand, and brought them near unto him.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#008888| And Israel stretched out his right hand, and laid it upon Ephraim's head, who was the younger, and his left hand upon Manasseh's head, guiding his hands wittingly; for Manasseh was the firstborn.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#008888| And he blessed Joseph, and said, God, before whom my fathers Abraham and Isaac did walk, the God which fed me all my life long unto this day,}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#008888| The Angel which redeemed me from all evil, bless the lads; and let my name be named on them, and the name of my fathers Abraham and Isaac; and let them grow into a multitude in the midst of the earth.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#008888| And when Joseph saw that his father laid his right hand upon the head of Ephraim, it displeased him: and he held up his father's hand, to remove it from Ephraim's head unto Manasseh's head.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#008888| And Joseph said unto his father, Not so, my father: for this is the firstborn; put thy right hand upon his head.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#008888| And his father refused, and said, I know it, my son, I know it: he also shall become a people, and he also shall be great: but truly his younger brother shall be greater than he, and his seed shall become a multitude of nations.}}<ref name=":52" /> {{font|size=smaller|color=#0000FF|20}}{{font|color=#888800| And he blessed them that day, saying, In thee shall Israel bless, }}<ref name=":52" /> {{font|color=#008888|saying, God make thee as Ephraim and as Manasseh: and he set Ephraim before Manasseh.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#008888| And Israel said unto Joseph, Behold, I die: but God shall be with you, and bring you again unto the land of your fathers.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#008888| Moreover I have given to thee one portion above thy brethren, which I took out of the hand of the Amorite with my sword and with my bow.}}<ref name=":52" /> ==Chapter 49== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Jacob called unto his sons, and said, Gather yourselves together, that I may tell you that which shall befall you in the last days.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| Gather yourselves together, and hear, ye sons of Jacob; and hearken unto Israel your father.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| Reuben, thou art my firstborn, my might, and the beginning of my strength, the excellency of dignity, and the excellency of power:}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| Unstable as water, thou shalt not excel; because thou wentest up to thy father's bed; then defiledst thou it: he went up to my couch.}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| Simeon and Levi are brethren; instruments of cruelty are in their habitations.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| O my soul, come not thou into their secret; unto their assembly, mine honour, be not thou united: for in their anger they slew a man, and in their selfwill they digged down a wall.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| Cursed be their anger, for it was fierce; and their wrath, for it was cruel: I will divide them in Jacob, and scatter them in Israel.}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| Judah, thou art he whom thy brethren shall praise: thy hand shall be in the neck of thine enemies; thy father's children shall bow down before thee.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| Judah is a lion's whelp: from the prey, my son, thou art gone up: he stooped down, he couched as a lion, and as an old lion; who shall rouse him up?}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| Binding his foal unto the vine, and his ass's colt unto the choice vine; he washed his garments in wine, and his clothes in the blood of grapes:}} {{font|size=smaller|color=#0000FF|12}}{{font|color=#000088| His eyes shall be red with wine, and his teeth white with milk.}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#000088| Zebulun shall dwell at the haven of the sea; and he shall be for an haven of ships; and his border shall be unto Zidon.}} {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| Issachar is a strong ass couching down between two burdens:}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And he saw that rest was good, and the land that it was pleasant; and bowed his shoulder to bear, and became a servant unto tribute.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| Dan shall judge his people, as one of the tribes of Israel.}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| Dan shall be a serpent by the way, an adder in the path, that biteth the horse heels, so that his rider shall fall backward.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| I have waited for thy salvation, O LORD.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| Gad, a troop shall overcome him: but he shall overcome at the last.}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| Out of Asher his bread shall be fat, and he shall yield royal dainties.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Naphtali is a hind let loose: he giveth goodly words.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| Joseph is a fruitful bough, even a fruitful bough by a well; whose branches run over the wall:}} {{font|size=smaller|color=#0000FF|23}}{{font|color=#000088| The archers have sorely grieved him, and shot at him, and hated him:}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#000088| But his bow abode in strength, and the arms of his hands were made strong by the hands of the mighty God of Jacob; (from thence is the shepherd, the stone of Israel:)}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#000088| Even by the God of thy father, who shall help thee; and by the Almighty, who shall bless thee with blessings of heaven above, blessings of the deep that lieth under, blessings of the breasts, and of the womb:}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#000088| The blessings of thy father have prevailed above the blessings of my progenitors unto the utmost bound of the everlasting hills: they shall be on the head of Joseph, and on the crown of the head of him that was separate from his brethren.}} {{font|size=smaller|color=#0000FF|27}}{{font|color=#000088| Benjamin shall ravin as a wolf: in the morning he shall devour the prey, and at night he shall divide the spoil.}} {{font|size=smaller|color=#0000FF|28}}{{font|color=#000088| All these are the twelve tribes of Israel: and this is it that their father spake unto them, and blessed them; every one according to his blessing he blessed them.}}<ref name=":53">{{Cite web|url=https://twitter.com/joelbaden/status/1325088868081790979|title=Genesis 49|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|size=smaller|color=#0000FF|29}}{{font|color=#888800| And he charged them, and said unto them, I am to be gathered unto my people: bury me with my fathers in the cave that is in the field of Ephron the Hittite,}} {{font|size=smaller|color=#0000FF|30}}{{font|color=#888800| In the cave that is in the field of Machpelah, which is before Mamre, in the land of Canaan, which Abraham bought with the field of Ephron the Hittite for a possession of a buryingplace.}} {{font|size=smaller|color=#0000FF|31}}{{font|color=#888800| There they buried Abraham and Sarah his wife; there they buried Isaac and Rebekah his wife; and there I buried Leah.}} {{font|size=smaller|color=#0000FF|32}}{{font|color=#888800| The purchase of the field and of the cave that is therein was from the children of Heth.}} {{font|size=smaller|color=#0000FF|33}}{{font|color=#888800| And when Jacob had made an end of commanding his sons, he gathered up his feet into the bed, and yielded up the ghost, and was gathered unto his people.}}<ref name=":53" /> ==Chapter 50== {{font|size=smaller|color=#0000FF|1}}{{font|color=#000088| And Joseph fell upon his father's face, and wept upon him, and kissed him.}} {{font|size=smaller|color=#0000FF|2}}{{font|color=#000088| And Joseph commanded his servants the physicians to embalm his father: and the physicians embalmed Israel.}} {{font|size=smaller|color=#0000FF|3}}{{font|color=#000088| And forty days were fulfilled for him; for so are fulfilled the days of those which are embalmed: and the Egyptians mourned for him threescore and ten days.}} {{font|size=smaller|color=#0000FF|4}}{{font|color=#000088| And when the days of his mourning were past, Joseph spake unto the house of Pharaoh, saying, If now I have found grace in your eyes, speak, I pray you, in the ears of Pharaoh, saying,}} {{font|size=smaller|color=#0000FF|5}}{{font|color=#000088| My father made me swear, saying, Lo, I die: in my grave which I have digged for me in the land of Canaan, there shalt thou bury me. Now therefore let me go up, I pray thee, and bury my father, and I will come again.}} {{font|size=smaller|color=#0000FF|6}}{{font|color=#000088| And Pharaoh said, Go up, and bury thy father, according as he made thee swear.}} {{font|size=smaller|color=#0000FF|7}}{{font|color=#000088| And Joseph went up to bury his father: and with him went up all the servants of Pharaoh, the elders of his house, and all the elders of the land of Egypt,}} {{font|size=smaller|color=#0000FF|8}}{{font|color=#000088| And all the house of Joseph, and his brethren, and his father's house: only their little ones, and their flocks, and their herds, they left in the land of Goshen.}} {{font|size=smaller|color=#0000FF|9}}{{font|color=#000088| And there went up with him both chariots and horsemen: and it was a very great company.}} {{font|size=smaller|color=#0000FF|10}}{{font|color=#000088| And they came to the threshingfloor of Atad, which is beyond Jordan, and there they mourned with a great and very sore lamentation: and he made a mourning for his father seven days.}} {{font|size=smaller|color=#0000FF|11}}{{font|color=#000088| And when the inhabitants of the land, the Canaanites, saw the mourning in the floor of Atad, they said, This is a grievous mourning to the Egyptians: wherefore the name of it was called Abelmizraim, which is beyond Jordan.}}<ref name=":54">{{Cite web|url=https://twitter.com/joelbaden/status/1325418395404275714|title=Genesis 50|author=Joel Baden|website=Twitter|language=en|access-date=2021-10-18}}</ref> {{font|size=smaller|color=#0000FF|12}}{{font|color=#888800| And his sons did unto him according as he commanded them:}} {{font|size=smaller|color=#0000FF|13}}{{font|color=#888800| For his sons carried him into the land of Canaan, and buried him in the cave of the field of Machpelah, which Abraham bought with the field for a possession of a buryingplace of Ephron the Hittite, before Mamre.}}<ref name=":54" /> {{font|size=smaller|color=#0000FF|14}}{{font|color=#000088| And Joseph returned into Egypt, he, and his brethren, and all that went up with him to bury his father, after he had buried his father.}} {{font|size=smaller|color=#0000FF|15}}{{font|color=#000088| And when Joseph's brethren saw that their father was dead, they said, Joseph will peradventure hate us, and will certainly requite us all the evil which we did unto him.}} {{font|size=smaller|color=#0000FF|16}}{{font|color=#000088| And they sent a messenger unto Joseph, saying, Thy father did command before he died, saying,}} {{font|size=smaller|color=#0000FF|17}}{{font|color=#000088| So shall ye say unto Joseph, Forgive, I pray thee now, the trespass of thy brethren, and their sin; for they did unto thee evil: and now, we pray thee, forgive the trespass of the servants of the God of thy father. And Joseph wept when they spake unto him.}} {{font|size=smaller|color=#0000FF|18}}{{font|color=#000088| And his brethren also went and fell down before his face; and they said, Behold, we be thy servants.}} {{font|size=smaller|color=#0000FF|19}}{{font|color=#000088| And Joseph said unto them, Fear not: for am I in the place of God?}} {{font|size=smaller|color=#0000FF|20}}{{font|color=#000088| But as for you, ye thought evil against me; but God meant it unto good, to bring to pass, as it is this day, to save much people alive.}} {{font|size=smaller|color=#0000FF|21}}{{font|color=#000088| Now therefore fear ye not: I will nourish you, and your little ones. And he comforted them, and spake kindly unto them.}} {{font|size=smaller|color=#0000FF|22}}{{font|color=#000088| And Joseph dwelt in Egypt, he, and his father's house: and Joseph lived an hundred and ten years.}}<ref name=":54" /> {{font|size=smaller|color=#0000FF|23}}{{font|color=#008888| And Joseph saw Ephraim's children of the third generation: the children also of Machir the son of Manasseh were brought up upon Joseph's knees.}} {{font|size=smaller|color=#0000FF|24}}{{font|color=#008888| And Joseph said unto his brethren, I die: and God will surely visit you, and bring you out of this land unto the land which he sware to Abraham, to Isaac, and to Jacob.}} {{font|size=smaller|color=#0000FF|25}}{{font|color=#008888| And Joseph took an oath of the children of Israel, saying, God will surely visit you, and ye shall carry up my bones from hence.}} {{font|size=smaller|color=#0000FF|26}}{{font|color=#008888| So Joseph died, being an hundred and ten years old: and they embalmed him, and he was put in a coffin in Egypt.}}<ref name=":54" /> == References == [[Category:Documentary hypothesis]] nv04qox6o81uzl4yhdjdp6q487nqtop 0 77105 2691721 2456868 2024-12-12T23:37:36Z Aerrapc 2951354 code formating 2691721 wikitext text/x-wiki '''LaTeX''' is a [[markup language]] (as is [[MediaWiki]]!) for producing mathematical texts of the highest quality. Its use is widespread in the mathematics world. It is built on plain [[w:TeX|TeX]] developed by [[W:Donald Knuth|Donald Knuth]]. You can embed LaTeX markups in MediaWiki by the <code><nowiki></math></nowiki></code> tags, e.g. <kbd><code><nowiki><math>e = m c^2</math></nowiki></code></kbd> is rendered as <math> e = m c^2</math>. == Readings == * [[Wikipedia: LaTeX]] * [[Wikibooks: LaTeX]] ==Installation== *If you are using Linux, you probably have it already! Just type "latex". *If you are using Windows, the simplest thing to do is to open [[w:cygwin|Cygwin]] (the linux emulator) and type "latex". If you haven't installed Cygwin, do it! You will often find it useful. The installation may take less than an hour but it is usually straightforward. ** Note that it is now fairly simple to install a [[linux dual boot]], for example, with [[wubi]]. All you need is a number of Gigabytes in a refactored hard-drive. You can delete it like a regular windows directory whenever you don't want it any more. ==Step by step guide for beginners== #Create a file with .tex suffix, e.g. <code>helloworld.tex</code> (see [[/helloworld.tex]]). #Go to your <code>linux/cygwin/</code><ref>If you are new to cygwin, you can find your c-drive by something like <kbd>cd /cygdrive/c</kbd> </ref> whatever shell, go to the directory of your file <code>helloworld.tex</code> and type <code>latex helloworld</code>. #A [[w:dvi|.dvi]] file will be created - you may view it directly if you have the right tools. #If not, type <code>dvips helloworld</code> or type <code>dvipdf helloworld</code> to convert it to a [[w:postscript|postscript]] or [[w:pdf|PDF]] file. #Use [[w:Ghostview|Ghostview]] to view the .ps file, or the [[w:Adobe Acrobat Reader|Adobe Acrobat Reader]] to view the .pdf file. ==Choosing an editor== A good editing environment helps! A useful and free option is the [[w:LaTeX-Editor_(LEd)|LaTeX editor (LEd)]] ([http://www.latexeditor.org/ see their home page]). A [http://ooolatex.sourceforge.net LaTeX extension] is also available for [[Open Office]]. Or for a quick and easy solution, try [http://rogercortesi.com/eqn/index.php Roger's Online Equation Editor]. == Lessons == * [[/helloworld.tex/]] * [[/Including graphics/]] * [[/Inserting diagrams/]] * [[/Two-lined-subscripts/]] * [[/Serial letter/]] ==Obtain PDF from Latex== pdflatex hello.tex == Use Latex Source in Wikiversity == You can use latex sources also in Wikiversity. In general you will create a PDF file from your latex source. <kbd>pdflatex</kbd> converts the a tex-file <kbd>hello.tex</kbd> in <kbd>hello.pdf</kbd>. You may need sometimes a different output format. If you want to convert the tex-file into a Wiki markdown output for Wikiversity you can use [https://pandoc.org/try PanDoc]. === Pandoc === This is useful if you want to use parts from source text in LaTeX also in a Wikiversity learning resource (e.g. for a mathematical text for a lecture). Copy the source in [https://pandoc.org/try PanDoc] and * select [https://pandoc.org/try "Latex" as input format] and * select [https://pandoc.org/try "MediaWiki" as an output format]. When press [https://pandoc.org/try "convert"] you can use the Wiki source code in your learning resource. === Expand Newcommands === Sometimes it is necessary to [https://niebert.github.io/closingbracket expand latex macros] because the macros in your source are not available in Wikiversity. * With the BracketHandler <kbd>closingbracket.js</kbd> a newcommand replacement into the source text is possible. * Use [https://niebert.github.io/closingbracket Newcommand Expander] as an [[AppLSAC]] for replacing the newcommand ([https://www.github.com/niebert/closingbracket GitHub Repository closingbracket.js]). ==External Links== * [http://tex.loria.fr/general/texbytopic.pdf tex by topic] - an indepth study of TeX * [https://niebert.github.io/closingbracket closingbracket.js BracketHandler] - GitHub demo for newcommand replacement with [https://www.github.com/niebert/closingbracket GitHub-repository 'closingbracket'] == References == <references/> [[category:LaTeX| ]] [[category:mathematics]] [[category:computer science]] [[category:academic publishing]] o76e0xq62vvfhitfj64tyxd6ofbmo0d 0 96254 2691645 2631735 2024-12-12T15:53:32Z DerGeist4040 2994919 /* The Post-Marian Legion */ 2691645 wikitext text/x-wiki This lesson will introduce you to the military terminology used in ancient Greece and Rome and will contain practical examples and scenarios so that if you happen to land in an ancient Roman or Greek battlefield, you'll know what to do. == The Greek Warfare == The Greeks are known to be the cradle of western civilization - they are the ones that gave us works such as Plato's the Republic, Homer's Odyssey and numerous other works. But they innovated a great deal in the subject of warfare as well, they created the phalanx formation; this was adapted to the [[w:Macedonian phalanx|Macedonian Phalanx]] which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain --everything that isn't a dry flat plain on a sunny day-- the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will reffer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a maniple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman stateman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own al ot of property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordeace to this he eliminated the previous system of a three line battlaion, instead he instituted ten cohorts composing each legion with additional nonc-ombatat troops, each cohort was composed of six centuriae of 80 men each,w ith each centuriybeing a unit in itself, with its own supplies and arms. Each centuriywas further divided into eight units of contubernia whos' only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears ,used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javlin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - "why should the Roman general use his greatste resource - the trairii only as the last resort and not as the first line who would probably crush the enemy?" ”he answer is that first, the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed and armoured soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a vetrera force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line ,then the other soldeirs would never become expierinced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training everyday and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken greed artillery pieces and improved them. One key character the Romans had was that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === but it was fun ==See also== [[Category:Military History]] [[Category:Ancient Greece]] cr1jw79bjjhis4jjjyy7srtw8v2a7je 2691648 2691645 2024-12-12T16:24:11Z DerGeist4040 2994919 /* The Greek Warfare */ 2691648 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain --everything that isn't a dry flat plain on a sunny day-- the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will reffer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a maniple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Roman general use his greatest resource - the trairii only as the last resort and not as the first line who would probably crush the enemy?” ”he answers is that first, the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed and armored soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training every day and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken greed artillery pieces and improved them. One key character of the Romans had been that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] p1taexhe2fyde3xlmx8eybxhgo6zn43 2691649 2691648 2024-12-12T16:26:28Z DerGeist4040 2994919 More grammar, spelling, and quality of life changes. 2691649 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain --everything that isn't a dry flat plain on a sunny day-- the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Roman general use his greatest resource - the trairii only as the last resort and not as the first line who would probably crush the enemy?” ”he answers is that first, the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed and armored soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training every day and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken greed artillery pieces and improved them. One key character of the Romans had been that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] bs0yrahplqbw3pm8z4bsjk0lz8bvm15 2691651 2691649 2024-12-12T16:27:54Z DerGeist4040 2994919 2691651 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Roman general use his greatest resource - the trairii only as the last resort and not as the first line who would probably crush the enemy?” ”he answers is that first, the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed and armored soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training every day and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken greed artillery pieces and improved them. One key character of the Romans had been that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] i327s9sc2wv0o8crp8m9qeibsoipbxy 2691655 2691651 2024-12-12T16:42:04Z DerGeist4040 2994919 /* The Qualities of the pre-Marian Roman Legion */ 2691655 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training every day and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. One key character of the Romans had been that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] du9buajidjus9iev68bv3qomenubg25 2691657 2691655 2024-12-12T16:49:29Z DerGeist4040 2994919 2691657 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. '''WARNING''': Some content of this page is incorrect. I advise you to use this resource at your own risk. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === The military is now compromised of professional soldiers training every day and building camps at night. The infantry classes now have similar equipment with equal quality. The infantry is trained to fight in formations and not as individual fighters. The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. One key character of the Romans had been that they didn't care about casualties as long as they won the war. They didn't send soldiers to die unnecessarily, but 1 soldier could kill two opposing soldiers with their pila, and remember that the pila went through both shields and armor, meaning even veteran soldiers of the enemy would be just as vulnerable as a raw recruit. The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] os02aju14spwgl3tfdb91nzy8js4ylf 2691659 2691657 2024-12-12T16:53:05Z DerGeist4040 2994919 Fixed some formatting 2691659 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. '''WARNING''': Some content of this page is incorrect. I advise you to use this resource at your own risk. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The First (and Second) Legion === Romulus - the founder of Rome, instituted the first legion in the Roman history - a legion composed of exactly 3,301 men - 3,000 infantrymen and 300 cavalry each third from each of the founding tribes of Rome. We are almost totally positive that this estimate is wrong. And yet we know that the first Roman legion had a few of the strategic qualities of the latter legions. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === There were many changes in the post-Marian Legion. Here are a few: * The military is now comprised of professional soldiers training every day and building camps at night. * The infantry classes now have similar equipment with equal quality. * The infantry is trained to fight in formations and not as individual fighters. * The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. * They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. * Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. * The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. * One key character of the Romans had been that they didn't care about casualties as long as they won the war. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] ihszex9yqnkdacckwgi9w6ghddgb0z8 2691680 2691659 2024-12-12T19:44:18Z DerGeist4040 2994919 I deleted this section mainly for two reasons. It says itself the estimate is wrong, and Romulus is a myth. 2691680 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. '''WARNING''': Some content of this page is incorrect. I advise you to use this resource at your own risk. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Romans grew from a village in rural Italy to an Empire, that we owe to as much as we owe the Greek, and perhaps even more. They grew for one reason - their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest "reliable" preliminary source is Livy - who was born in 59 BC. This does not apply to Greeks before 59 BC. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === There were many changes in the post-Marian Legion. Here are a few: * The military is now comprised of professional soldiers training every day and building camps at night. * The infantry classes now have similar equipment with equal quality. * The infantry is trained to fight in formations and not as individual fighters. * The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. * They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. * Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. * The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. * One key character of the Romans had been that they didn't care about casualties as long as they won the war. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] i4hc6pl1iyhecvwx63b0fmjboa2cjxg 2691681 2691680 2024-12-12T19:50:58Z DerGeist4040 2994919 /* The Roman Warfare */ Linked the page for livy 2691681 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. '''WARNING''': Some content of this page is incorrect. I advise you to use this resource at your own risk. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Roman Empire grew from a village in rural Italy to one of the greatest empires. They grew for many reasons. One such reason is; their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest reliable preliminary source is [[wikipedia:Livy|Titus Livius II]] - who was born in 59 BC. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of what I will refer to as the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and accordance to this he eliminated the previous system of a three line battalion, instead he instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === There were many changes in the post-Marian Legion. Here are a few: * The military is now comprised of professional soldiers training every day and building camps at night. * The infantry classes now have similar equipment with equal quality. * The infantry is trained to fight in formations and not as individual fighters. * The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. * They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. * Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. * The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. * One key character of the Romans had been that they didn't care about casualties as long as they won the war. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] 2a1uzr8d5m75qt0mq355b4tsxw7uoz8 2691683 2691681 2024-12-12T20:02:46Z DerGeist4040 2994919 /* The Pre-Marian Legion */ 2691683 wikitext text/x-wiki This lesson will provide an introduction to the military terminology utilized in Ancient Greece and Rome. It will also include practical examples and scenarios, enabling you to be prepared in the event of encountering an ancient Roman or Greek battlefield. '''WARNING''': Some content of this page is incorrect. I advise you to use this resource at your own risk. == The Greek Warfare == Although the Greeks are known for being the cradle of western civilization, they also innovated greatly in the subject of warfare. They created the phalanx formation, which was adapted to the Macedonian Phalanx, which Alexander the Great used to conquer the previously superior Persian Empire. === The Hoplite Phalanx === The classical phalanx formation is composed of usually 8 ranks of varying numbers of men, some phalanxes were even deeper - up to 50 ranks in an exceptional battle. The phalanx formation was designed the following way: the soldiers were held closely together, each holding the shield before him and slightly to the left. The result was a heavy juggernaut, a mass of tightly packed soldiers all marching, totally shielded from their opponents by both their shields and their fellow soldiers shields. Each were equipped with long spears protruding 2–7 yards before them. In addition, not only the first line projected its spears forward, but the next ones as well, each in a slightly bigger angle which made the formation even more formidable. And yet, there were two great disadvantages, first; such a formation could crush anything that it met with straight ahead of it, but not anything that attacked it from the flank, or rear. When they marched on an unstable terrain, the close formation exercised during battle and the cumbersome spears that made it practically impossible to change the direction the formation faced, even if one soldier lost his footing on a stone the whole formation would be out of order and most likely fail. The second disadvantage was that the Greek city-states who used the phalanx loved to fight between themselves, so in almost all cases, the phalanx might be fearsome, but the enemy had the exact same ideas as well as training. == The Roman Warfare == The Roman Empire grew from a village in rural Italy to one of the greatest empires. They grew for many reasons. One such reason is; their military. And this will be examined in the coming sections. Remember though - the Gauls sacked Rome in 400 BC, and so we have no preliminary sources for anything before that time, which is the Roman kingdom and the early Roman, and even so the earliest reliable preliminary source is [[wikipedia:Livy|Titus Livius II]] - who was born in 59 BC. === The Pre-Marian Legion === After the sack of Rome by the Gauls, who was preceeded by a distarous defeat of the entire Roman army against the Gauls in Allia, led to the creation of the pre-Marian legion, a legion composed of three lines of solieirs with varying quality and of support troops, called a manciple. The maniple was lied out in a checkerboard formation designed to easily traverse the hilly landscape of Samnia. === The Marian reforms === Gaius Marius was a Roman statesman in the 1st century BC, he was the one who as a consul (there were two consuls, each elected for a term of a year who were the highest authority in Rome)totally changed the face of the Roman army - before him, a soldier had to own property, be of a high class and supply his own weapons and armor, only to qualify for a soldier. Marius allowed the landless masses to become soldeirs, and as this option offered them a permanent pay and a modest living, many joined the army. In addition and in accordance to this, he eliminated the previous system of a three line battalion. He instituted ten cohorts composing each legion with additional noncombatant troops, each cohort was composed of six centuries of 80 men each, with each century being a unit in itself, with its own supplies and arms. Each century was further divided into eight units of contubernia who's only function was that each contubernia slept in the same tent. The first line in the Marian legion was made up of Hastati. Hastati were the youngest men in the army and therefore having the most stamina, which was perfect for a front line. Hastati carried two throwing spears or Hasta, a Gladius, a dagger, and a heavy shield or Scutum. The second line in the Marian legion was made of Principes. Principes were essentially Hastati, except they were the veterans of the battlefield. The third line in the Marian legion consisted of Triarii. Triarii were the closest Rome would come to a Phalanx. Triarii carried long spears, used effectively against all enemy units. The fourth line in the Marian legion was made up of Velites. Velites carried many small throwing spears and would rain volleys of spears down on the enemy. The last line of Marius' legion was the cavalry. They were called Equites because of Romulus' original army. In Romulus' time, Equite was the name for the richest people in Rome. Only the rich could afford horses because back then, everyone had to buy their own weapons and armor. == The Quality of the Roman Legion == === The Qualities of the pre-Marian Roman Legion === The early Roman army was composed entirely of citizens, some without any experience and almost no equipment, and some soldiers with the best equipment in Rome and years and years of fighting behind them. The citizens were organized at first into six tribunes who served under the general, usually the king. Most soldiers were javelin-throwers, others were archers. The legionaries were arranged into three lines, with the addition of cavalry companies (equites, coming from the Latin word for horse - Equus) and velites - light troops who were used to skirmish the enemy and screen the main line. The three main lines of the legion were (from youngest to most experienced)- Hastati, principes and Triarii. The Hastati would be the first line, advancing after the velites attached to them harassed the enemy and retreated, they would engage the enemy and if they failed to break it they would retreat behind the principes line who would then advance on the enemy, if it failed the triarii would advance forward, as the last resort. Behind the triarii the rorarii and accensi stood, the poorest troops, armed only with slings who were used in a support role mainly and probably either never saw battle or were used as one-time cannon fodder. The most obvious question about this tactic is - “why should the Legate use his greatest resource, the trairii, only as the last resort and not as the first line?” the answer is: the triarii, if used, would face an enemy which no matter how superior it was has by now faced two waves of javlines, two waves of charging sufficiently armed soldiers, one of which is experienced, and constantly being flanked by units of cavalry, any force would then fall to a veteran force using the best equipment Rome had to offer. Second, if the trairii would always fight as the first line, then the other soldeirs would never become experienced enough to be the next generation of the trairii. === The Post-Marian Legion === There were many changes in the post-Marian Legion. Here are a few: * The military is now comprised of professional soldiers training every day and building camps at night. * The infantry classes now have similar equipment with equal quality. * The infantry is trained to fight in formations and not as individual fighters. * The Romans feared the forest as the Romans needed time to deploy into formations and a sudden attack on the Romans could potentiality lead to a massacre. * They were very vulnerable on the march and so relied on allied cavalry to scout the surroundings and spies to get knowledge of the villages in the surroundings on the public opinion of the Romans. * Auxiliary troops archers slingers light infantry cavalry would come from non Romans from people that specialized in other traditions than the roman infantry core, this made the roman war machine extremely formidable as it had almost no weakness. * The Romans knew how to siege cities, they had taken artillery pieces and improved them and improved their tactics. * One key character of the Romans had been that they didn't care about casualties as long as they won the war. === The Late Imperial Legion === === The Roman Military Tradition === ==See also== [[Category:Military History]] [[Category:Ancient Greece]] bcqc7z3tln03skhje0k0r0kwano6x2g 0 99534 2691619 2060849 2024-12-12T13:22:11Z 27.147.228.7 /* Planning Stage */ 2691619 wikitext text/x-wiki This is offered as a checklist from which you, the event organiser, can select points that should be raised in staff planning meetings and used for ‘ticking off’ as part of your thorough preparation. Not all items listed will be part of all events of course! A number of items may appear more than once or may be noted in one section but you may wish to consider them for another section of the event plan. Items are not necessarily in the order they would be considered or in order of importance. You will select the relevant checkpoints as this is an organic collection of checkpoints and will change as others are added or some are modified! ==Planning Stage== # Preliminary research and feasibility study, including SWOT analysis # Event goals #Setting up the project management group (e.g. the event, define event project & scope, feasibility study, planning needs, staffing, site, monitoring, quality control, planning schedules) #Economic impact #Budget: economic analysis; pricing; contingencies; break-even point; sources of funding; account codes & prefixes; cash-flow analysis; money needed onsite; #Site selection: facilities; location; access to accommodation; shops and transport #Event theme & branding #Analyse venue in terms of consumer needs #Flow charts of planning decisions and actions #Operational policies and procedures manual #Logistics #Employment law relevant to workers on the event day #Staffing and recruitment of staff, training, job descriptions, accreditation #Employee contracts #Volunteers: their roles and training #Uniforms, caps or shirts for staff #Consideration of any outsourcing #Guest list and invitations #Publicity and media #Action Plans for individuals or groups #Legal compliance, including permits and licences <ref>{{Cite web|url=https://www.icslegal.co/|title=Apply for a Sponsor Licence {{!}} Step-by-Step Guidance UK & US|website=Sponsor Management System|language=en|access-date=2024-12-12}}</ref>and compliance from toilets to health rules to power cables #Projections of attendance & participation #Community consultation e.g. traffic, on possible noise #Prizes, awards and lucky tickets or lucky seats #Sponsorship #Seating capacities and for whom and location #Fences & perimeters #Signage & advertising #Ticketing #Concessions, passes #Naming rights #Services: power; water, gas; emergency generator; lighting; refrigeration #Decorations, posters and signs #Photography #Maps and plans #Sale points and merchandising #Stalls with their operation and location #Media: contacts; pre-event articles; features to publicise; use of social media; media persons at the event (space, facilities, location, food and drink, interim updates, power; interviews; event follow-up) #Special needs: wheelchair access; ramps; location of wheelchair viewing sites; toilets for disabled; parking for disabled) #Exhibition space #ATM machines #Hospitality – for whom and location of hospitality sites #Noise levels #Ground announcer and the provision of key scripts for the announcer (e.g. sponsors, event happenings; schedule, lost children, lost & found) #Opening ceremony ==Marketing== #Marketing strategy and marketing plan #Forming a marketing team #IMC approach may be appropriate (the definitions vary somewhat but IMC or Integrated Marketing Communications may be a plan that links all promotion modes and has contacts and communications consistent in their relation to your brand with consistent meanings expressed and reinforced) #Sponsorship: determining likely sponsors; obtaining sponsorship; providing for sponsor publicity before/at/after the event #Promotional materials #Utilise the organisation’s branches or sport club affiliates #Signage #Clothing with sponsor or event organisation name or insignia #Ambush marketing prevention #Use of a celebrity for event endorsement or opening the event #Media launch of the event #Trademarks, copyright materials and logos checked #Direct mail #Local media, national media #Merchandising #Onsite photocopying & printing #Marketing to enhance participant, spectator, visitor, reader/viewer experiences #Marketing to place (or establish) the event in the annual programme ==The Event Day and Event Environment== #Site maps including site-flow maps #Clear delineation of duties #Operations manual #Rehearsals #Published programme #Stage and/or designated areas #Cultural protocols to be observed #Insurance #Functional areas #Temperature control, provision for rain and adverse weather #Admission, queue prevention & access for wheelchairs and baby prams #Pass-out system #Risk Management policy and processes in place and checked on the event day: risk assessment; hazard analysis; security plans; police; emergency services contacts; locations of emergency facilities; first aid services; child safety; crisis response preparedness; smoking and/or drugs ban; access for emergency vehicles; duty of care; occupational health and safety; playing surfaces; crowd control; marshals; training in health and safety for staff; childcare #Food and Catering: food storage onsite; alcohol provision, access and regulation; caterers; food stalls; waste and recycling; food handling; cleaning; food preparation & service. #Lost property site and information about found objects #Lost children & meeting points #Sanitation and toilets: location; coping with emergencies; toilets for the disabled; signs showing toilet locations; portable toilet removal #Waste and recycling: bins; contracts; disposal. #Storage #Spectator comfort and visibility of event #Clearly designated seating #Entrances & exits #Parking and transport, including disabled persons parking and access to site #Public address system #Lighting #Special provision for participants checked: changing rooms; hot water; showers; support staff facilities; space; food & drink; after-match function; school-age participants could be congratulated by a known sport or media personality; awards; first-aid; stewards #Data collection to assist post-event analysis ==Post -Event Action== #Evaluation in terms of plans, event goals, budget #Venue evaluation in terms of suitability and ease of functions #Clearing, cleaning and dismantling – was this smoothly done? #Cleaning and clearance of waste #Employee satisfaction #Participant/spectator satisfaction #De-briefing meetings of staff #Sponsorship evaluation #Thanks to volunteers or a function for them #What improvements will be made for next time? 4u3cvyijfa47xzts21obh86qq5846is 0 113381 2691663 2691512 2024-12-12T17:04:55Z Eshaa2024 2993595 2691663 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> have a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]], *'''[[Holomorphism|Holomorphism]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism&author=course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Holomorphism&coursetitle=Course:Complex_Analysis Slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Course:Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[/Inequalities/]] **[[Complex Analysis/rectifiable curve|rectifiable curve]] * '''[[ Course:Complex Analysis/Curve|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=%20Course:Complex%20Analysis/Curve&author=%20Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve&coursetitle=%20Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Curve%20Integral&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Path%20of%20Integration&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat%27s%20Lemma%20(Details)&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course%3A+Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&author=Course%3A+Complex+Analysis&audioslide=yes&language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]], * '''[[Course: Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Identity%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course: Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Liouville%27s%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville%27s%20Theorem&coursetitle=Course:%20Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> 3r2nunaaupugq7p0ccipn8g1vpaqnbk 2691740 2691663 2024-12-13T07:25:11Z Bert Niehaus 2387134 /* Chapter 4 - Curves and Line Integrals */ inappropriate duplicate translation with a different page header. 2691740 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> have a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]], *'''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic_function&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Holomorphic function&coursetitle=Complex_Analysis Slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Course:Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[/Inequalities/]] **[[Complex Analysis/rectifiable curve|rectifiable curve]] * '''[[ Course:Complex Analysis/Curve|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=%20Course:Complex%20Analysis/Curve&author=%20Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve&coursetitle=%20Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Curve%20Integral&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Path%20of%20Integration&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat%27s%20Lemma%20(Details)&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course%3A+Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&author=Course%3A+Complex+Analysis&audioslide=yes&language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]], * '''[[Course: Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Identity%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course: Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Liouville%27s%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville%27s%20Theorem&coursetitle=Course:%20Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> cny8zz440syy8zgrycm1k3gnmmy2hxk 2691741 2691740 2024-12-13T07:27:20Z Bert Niehaus 2387134 /* Chapter 4 - Curves and Line Integrals */ 2691741 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> have a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]], * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Course:Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[/Inequalities/]] **[[Complex Analysis/rectifiable curve|rectifiable curve]] * '''[[ Course:Complex Analysis/Curve|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=%20Course:Complex%20Analysis/Curve&author=%20Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve&coursetitle=%20Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Curve%20Integral&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Path%20of%20Integration&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat%27s%20Lemma%20(Details)&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course%3A+Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&author=Course%3A+Complex+Analysis&audioslide=yes&language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]], * '''[[Course: Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Identity%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course: Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Liouville%27s%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville%27s%20Theorem&coursetitle=Course:%20Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> r3m6161yzyr1u1clxmqs2wze18l73k4 2691742 2691741 2024-12-13T07:27:56Z Bert Niehaus 2387134 /* Chapter 4 - Curves and Line Integrals */ 2691742 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> have a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]], * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Course:Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[/Inequalities/]] **[[Complex Analysis/rectifiable curve|rectifiable curve]] * '''[[ Course:Complex Analysis/Curve|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=%20Course:Complex%20Analysis/Curve&author=%20Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve&coursetitle=%20Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Curve%20Integral&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Path%20of%20Integration&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course:Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&author=Course:Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat%27s%20Lemma%20(Details)&coursetitle=Course:Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course%3A+Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&author=Course%3A+Complex+Analysis&audioslide=yes&language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]], * '''[[Course: Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Identity%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Course: Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:%20Complex%20Analysis/Liouville%27s%20Theorem&author=Course:%20Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville%27s%20Theorem&coursetitle=Course:%20Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> l5uxmyq4tnlv26rafxuffvxyiabjq3s 0 139384 2691615 2691402 2024-12-12T12:59:07Z Young1lim 21186 /* Adder */ 2691615 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]|| || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20241212.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 3dk9dzpavmz68lfmblxjvg49jqhyvc0 0 149103 2691618 1876017 2024-12-12T13:20:51Z 27.147.228.7 /* Flexibility in IP policy and procedures */ 2691618 wikitext text/x-wiki [[File:OpenEdRecommendationsforLTU-310513.pdf|thumb|300px|Discussion paper on open education, presented by the La Trobe Open Education Working Group to the Pro-Vice Chancellor for Future Learning.]] {{center top}}<big>'''Open education discussion paper'''</big>{{center bottom}}<br> {{center top}}Open Education Working Group, La Trobe University May 2013{{center bottom}}<br> This document was produced by the Open Education Working Group at La Trobe University in response to the [http://www.radicallearninggroup.blogspot.com.au/ Radical Learning Group] findings and recommendations. The working group consists of representatives across three faculties, the library and central teaching and learning unit. Members are: Donna Bisset (Humanities and Social Sciences), Leigh Blackall (Health Sciences), John Hannon (Curriculum, Teaching and Learning Centre), Simon Huggard (Digital Infrastructure), Ruth Jelley (Business, Economics and Law), Mungo Jones (Humanities and Social Sciences), Annabel Orchard (Business, Economics and Law), Roderick Sadler (Digital Infrastructure), Emily Krisenthal (Curriculum, Teaching and Learning Centre). The working group convened a one-day conference at La Trobe University on 12 March 2013 in order to gauge interest, experience and capability around open education. The event was attended by more than 50 teaching, administration and support staff as well as external guests. The planning and outcomes of that conference were thoroughly documented, and the findings inform this paper and are also summarised in Appendix 1. We present this discussion paper to encourage the La Trobe University to develop capacity and capability for open educational resources and practices within the time frame of the current strategic plan. Open education contributes to measurable improvement to learning outcomes and teaching efficacy. The increasing clarity of agenda in state and federal public service agencies, along with significant international trends in educational institutions suggests that it would be prudent for La Trobe University to invest in the development of open educational practices. La Trobe has the opportunity to develop policies that enable and encourage openness in educational practices and remove barriers (these are explicitly discussed in UNESCO, see Joyce 2006; Bossu et al 2012; D'Antoni 2009; Rolfe 2012). Policies and procedures that provide recognition, resources and support for open educational practices will support and contribute to the university's research culture of knowledge-sharing and knowledge-creation for staff and students. == Defining open education == Open education does not simply refer to access to freely available content; it includes non-restrictive terms and conditions and transparent development processes. Open education practices use collaborative methods and frameworks to develop content, curriculum, assessment, research, policies, projects, budgets and so on. The word ''open'' means unrestricted online access, radical transparency, maximum reusability and participatory collaboration. This meaning is informed by the principles and methods of free software espoused by w:Richard Stallman|Richard Stallman since 1983 (Williams 2002). These principles were first appropriated into educational contexts by David Wiley in 1998 (Wiley; Gurrell 2009), and have continued to grow. Hence, not all accessible content is open in this sense and there are limitations on (re)usability and transparency (Wiley 2012). Openness and open education and research practices now hold political significance, attract development funding and enjoy wide international infrastructure and volunteer support and guidance (The Interview 2012; McGreal 2012). === Open education characteristics === Open education has the following characteristics: ==== Open educational resources ==== # Public online teaching content that is freely licensed and archived in open standard formats. # Production processes that use and contribute to open educational resource standards. # Development of resources and processes aimed at reducing costs to students and improving educational outcomes. ==== Open research and education practices ==== # Publicly transparent planning, documentation and reporting of research, teaching and assessment. # Public online access to publications of research outcomes and the data that informs it. # Recognised and rewarded engagement with communities of open research and open education resource production and practice. ==== Transparent governance and open data ==== # Public online access to planning, reviews, reports, policies and procedures. # Open participatory collaboration in reviews, reports, deliberation and planning. # Public online documentation of governing procedures. == Adopting open education practices == The Radical Learning Group recommends that La Trobe engage with contemporary professional and pedagogical practices that are based on open access, participation, transparency and verifiable quality assurances. These practices are in keeping with La Trobe's objectives of the advancement of knowledge for the public good, and the Radical Learning Project report that calls for the university to 'position itself as an adaptive, innovative and creative institution' (Radical Learning Group 2012, p.3). Open education approaches offer La Trobe University the opportunity to improve teaching practices. Adopting open education practices involves a wide range of people and functions across the university, including library, faculty and central teaching and learning staff. === Improved educational outcomes === Adopting open educational practices will help improve student choices, experience, retention and completion, as well as the likelihood of return to study (Wiley 2013). ==== Reducing cost to students increases enrollment and completion ==== Wiley (2013) provides the example of Florida Virtual Campus, in which a large survey of students in 2012 revealed that the cost of textbooks was having a significant impact on student retention, completion of studies and academic performance. According to the study, 64% elected not to buy their textbooks due to cost, 30% decided not to enroll due to costs, 14% dropped courses/subjects due to the high cost of textbooks. Developing capacity to use open educational resources will reduce the cost of course materials and thus improve the student experience by reducing barriers to education. This goal and its outcomes align to Future Ready strategies to provide a high-quality student experience and nurture a diverse student body, including students from disadvantaged backgrounds (La Trobe University 2012, p. 5). ==== Continued access to up-to-date content improves lifelong professional learning ==== According to the Florida study, 70% of the 20 000 students surveyed valued having lifetime access to always up-to-date textbooks as important to very important. In a report on an OECD forum about OER, Joyce (2006) outlines the benefits of adopting an institutional approach to open education practices. Reporting on the forum, Joyce found that: Participants reflected on these and suggested a number of reasons for engaging in OER production at the institutional level, based upon issues such as continuing education for alumni, student course selection, attracting future students, cost reduction, alternatives to commercial materials, quality enhancement, interaction with a wider public, encouraging innovation, moral and ethical concerns, and legal requirements (page 8). This supports the case that the use of OERs at La Trobe can help establish and maintain ongoing contact with graduates, currently enrolled students, future students and professional associations. ==== Open access and transparency gives people a more informed choice ==== McAndrew et al (2010) note that the large increase in demand for higher education witnessed from the late 20th century onwards coincided with a wide proliferation of information and communication technologies that have enabled new models of education. Pedagogical models that provide transparency offer alternative modes of learning to the traditional, individualised teacher-student focus. These include open forms of assessment, authentic learning and student-generated knowledge. For example, a pedagogical model that provides transparency about learning objectives and progression is the Open University&rsquo;s Supported Open Learning Model, which empowers learners and has proven to be effective (Conole 2013). Baltzersen (2010) also discusses the pedagogical potential for transparent learning communities. ==== Closing the gap between formal and informal learning ==== Open education practices can provide the opportunity for universities to close the gap between formal and informal learning by moving education practice into informal learning environments and learning communities (Kanwar, Kodhandaraman &amp; Umar 2010). The flexibility that open educational resources and practices bring to the learning process help give value to informal learning (McAndrew et al 2010). Using open educational practices and resources can promote and enable lifelong learning. === Building capacity for OER === Building capacity for open education involves strategies for staff development, raising awareness and developing support structures for the adoption and creation of open educational resources. By enabling greater access to the university&rsquo;s teaching resources and encouraging academic staff to incorporate open resources into their teaching, open educational practices can improve quality and foster innovation in education (Bossu, Brown, Bull 2012). Openness is a key trend in education and La Trobe needs to keep pace with this trend if it is to adopt the '21st century-focused teaching and learning models' recommended in the Radical Learning Group report (2012). The 2013 Horizon Report on higher education discusses the importance of the trend towards openness in education (Johnson et al 2013). The Horizon Report identifies openness as a value; it is becoming a way of doing things across the university sector. Conole (2013) discusses the importance of adopting open education practices in modern curriculum design and teaching practice. The potential for OER to increase student engagement presents an opportunity to evaluate the effect of such practices on student retention. Rolfe (2012) notes the importance of collaboration among colleagues and the provision of technical assistance. Bueno-de-la-Fuente et al (2012) highlight the significance of library staff in the 'description and classification, management, preservation, dissemination, and promotion of OER' (p. 7). ==== Content up-to-date, peer reviewed, accessible and reusable ==== Publishing reusable digital content and artefacts online can reduce the time and cost associated with traditional publishing, as well as revision and republishing of teaching materials and resources (Baraniuk &amp; Burrus 2008, p.31). This would allow La Trobe University to provide relevant and useful content to current students, as well as alumni and professional associations in the wider community. ==== Copyright diligence and open standard formats ==== Engaging in the use and reuse of open educational resources necessitates substantial understanding of copyright and open standard formats. One of the main topics of discussion at the open education conference at La Trobe University (see Appendix 1) was that there existed inconsistent levels of understanding and use of resources, including unacknowledged and unapproved uses. The university can contribute to addressing this issue through engagement with open educational resources and practices. ==== Use of content without expensive contract negotiations and maintenance ==== The ability of faculties to source and use educational content and resources without the need to negotiate, enter into and maintain contracts with third-party providers presents a significant cost saving opportunity. This goes for content that is used by university staff as well as content produced by university staff. The university and its staff will need to understand their rights and responsibilities regarding the use and production of open educational resources. ==== Connecting the learning management system (LMS) to OER ==== Building capacity for OER requires developing interoperability of existing systems. One example is setting up a search and discovery environment for OER. This may involve reconfiguration and redevelopment of existing systems such as the LMS, Library eRepository, CMS and Intranet, as well as other systems. An approach that La Trobe could adopt is to make a connection between the learning management system (LMS) and OER. This would reconfigure the LMS as a gateway between the university and knowledge in open and collaborative online environments. This may involve packages or plugins that facilitate the automatic upload of OER generated by La Trobe staff into the La Trobe Library eRepository and external OER systems. Automatic upload of OERs to the Library eRepository will provide a consistent internal (and open) archival environment for materials produced by La Trobe University. Establishing these connections enables the university to recognise students&rsquo; existing use of social media for learning. Designing systems that 'prioritize the creation of identities for students at third-party sites, rather than bringing those sites within the learning system' (Allen &amp; Long 2010) will extend learning at La Trobe beyond centrally managed enterprise systems. This will support new learning approaches and pedagogies, following the Radical Learning Group recommendation 2. === Consistency with national and international developments === The trend towards open content is growing in strength and reach. Openness has been mandated in research and development funding in the UK, Australia, New Zealand, Canada and the USA. Openness is extending into government, universities in Australia are starting to engage in OER. ==== Government-supported open education and data initiatives ==== The Office of Teaching and Learning is providing funding for research into open education in Australia through projects such as the 'Adoption, use and management of OER to enhance teaching and learning in Australia' project (Bossu, Brown &amp; Bull, forthcoming). Moves that La Trobe makes into open educational practices will be in line with federal governmentinitiatives. Evaluation of any moves into open education can contribute much-needed evidence on the impact of OER on teaching and learning. Bossu et al&rsquo;s OLT-funded project has the potential to guide cross-institutional development of OER and connect interested parties. ==== Australian public service copyright benchmark ==== The Department of Finance and Deregulation and the Attorney General&rsquo;s Office now require public service agencies to make information accessible and reusable online, assigning the Creative Commons Attribution copyright license (Australian Government Intellectual Property Rules). This signifies a wider move towards open standards for publicly funded projects. ==== Australian university intellectual property (IP) policies that respect individual ownership and promote Creative Commons licensing ==== The Australian Council for University Librarians&rsquo; overview of university policies on intellectual property shows that a number of universities are already engaging in open educational practices. A range of Australian universities allow staff and students to retain ownership of their IP while working or studying at their university, such as University of New South Wales, University of Technology Sydney, University of Western Australia, University of Wollongong, James Cook University and Australian National University. The Australian National University is expressly supporting the use of Creative Commons copyright licensing. La Trobe University's IP policy statement in support of making teaching and learning materials publicly and freely available is the most progressive position in Australia, in that it supports individual ownership of IP and promotes academics to adopt open educational practices by allowing them greater autonomy over the copyright licensing of their work. Of the range of Creative Commons copyright license available, only two align to open education: the Attribution license and the Share Alike license. ==== Australian universities are actively engaged in open education ==== The University of Southern Queensland and the University of Wollongong are both founding anchors of the Open Educational Resources University (University of Southern Queensland 2012). The University of Tasmania is sponsoring the Adapt project to share teaching materials and develop teaching and scholarship. These activities are part of a wider movement that includes international organisations such as the Higher Education Academy (UK) and Open Courseware Consortium, some of which have received substantial funding from groups the likes of the William and Flora Hewlett Foundation. Removing restrictions on access to learning content and communities has enabled more people to engage in collaborative online learning. As other institutions adopt stronger policies of openness, adopting open practices ensures structures for effective collaboration with these organisations. In order to keep pace with this trend and demand and to take a leading role in its development, La Trobe University should begin investment in the capacity to build effective practices. == Recommendations == The Open Education Working Group makes the following recommendations regarding open educational practices at La Trobe University. === Development of policies and practices to support OER === Develop policies and practices to support institutional and individual capacity to produce, source, curate and use open educational resources in teaching and learning. Policies and practices to develop institutional capacity to support OER may encompass the library, information technology, media production, marketing, and faculty and central teaching and learning staff. The university will need to allocate resources for the development of open education practices and professional development programs to enable the production and (re)use of OER. In doing this, La Trobe can foster a culture that is wholly supportive of open education practices. Possible ways to do this include developing procedures to recognise OER engagement and creation. === Review of intellectual property (IP) policy === Review the La Trobe IP Policy to reduce barriers to open education practices. The current intellectual property policy was due for review in 2011. Revising the policy with a focus on open education practices will help clarify the university's position on open copyright licence options and the circumstances under which staff may elect to opt out of such processes. Ideally, a separate policy should be created for teaching and learning materials which clearly outlines what constitutes 'teaching and learning materials' and distinguishes them from exploitable IP. Revising the policy provides an opportunity to clarify whether attribution for teaching and learning materials should remain with individuals or the university. This revision can be accompanied by guidance for staff on making their materials available with a recommended Creative Commons licence. === Integrate open education practices with teaching and learning practices === Incorporate open education practices into the learning environment. The integration of open education practices into the learning environment aligns with the Radical Learning Project recommendation 2. This calls for the extension of teaching approaches beyond institutional boundaries and creation of opportunities for collaborative and experiential learning, as discussed in 3.2.4. === Partnerships === Explore strategic partnerships with external agencies and platforms to help build an open educational practice community at La Trobe University. Engagement with external projects and initiatives will assist La Trobe to implement effective policies and practices. These may include partnerships with universities or cross-organisational projects to develop national discipline-based OER. Participation in strategic and collaborative projects will develop sustainable knowledge resources with relevance and currency, and with potential impact for the university. == References == Allen, M., Long, J. (2009) Learning as knowledge networking: conceptual foundations for revised uses of the Internet in high education, Proceedings of the World Congress on Engineering and Computer Science, fromhttp://www.iaeng.org/publication/WCECS2009/WCECS2009_pp652-657.pdf Baltzersen, R. (2010) Radical transparency: Open access as a key concept in wiki pedagogy. Australasian Journal of Educational Technology 26(6), 791-809. http://www.ascilite.org.au/ajet/ajet26/baltzersen.html Baranuik, R., Burrus, C. (2008) Viewpoint: Global Warming Toward Open Educational Resources, Communications of the ACM, 51.9, 30-32. Bossu, C., Brown, M., &amp; Bull, D. (forthcoming) Feasibility Protocol: An instrument to assist institutional adoption of OER. Australian Government Office of Learning &amp; Teaching.http://wikiresearcher.org/OER_in_Australia/Planning Bossu, C., Brown, M., &amp; Bull, D. (2012). Do Open Educational Resrouces represent assitional challenges or advantages to the current climate of change in the Australian higher education sector, Proceedings Ascilite 2012 Conference, Retrieved from http://www.ascilite.org.au/conferences/wellington12/2012/images/custom/bossu%2c_carina_-_do_open.pdf Bueno-de-la-Fuente, G., Robertson, R.J., Boon, S. (2012) The roles of libraries and information professionals in Open Educational Resources (OER) initiatives: Survey Report. CAPLE/JISC CETIS. http://publications.cetis.ac.uk/2012/492 Conole, G. (2013) Designing for Learning in an Open World. New York: Springer. D&rsquo;Antoni, S (2009). Open Educational Resources: reviewing initiatives and issues, Open Learning: The Journal of Open, Distance and e-Learning, 24:1, 3-10 Florida Virtual Campus. (2012) 2012 Florida Student Textbook Survey. Tallahassee, Florida. http://www.openaccesstextbooks.org/pdf/2012_Exec_Sum_Student_Txtbk_Survey.pdf The Interview. (2012). Sir John Daniel, former President and CEO, Commonwealth of Learning. France 24. http://www.france24.com/en/20120712-john-daniel-open-educational-resources-movement last accessed 10 May 2013. Johnson, L., Adams Becker, S., Cummins, M., Estrada, V., Freeman, A. and Ludgate, H. (2013) NMC Horizon Report: 2013 Higher Education Edition. Austin, Texas: The New Media Consortium. Joyce, A. (2006) OECD study of OER: Forum Report Internet discussion forum: Open educational resources &ndash; Findings from an OECD study, UNESCO. Kanwar, A., Kodhandaraman, B., &amp; Unmar, A. (2010). Toward Sustainable Open Education Resources: A Perspective from the Global South, American Journal of Distance Education, 24 (2), pp 65-80. doi:10.1080/08923641003696588 La Trobe University (2012). Future Ready: Strategic Plan 2013-2017. Littlejohn, A., Beetham, H. and McGill, L. (2012). Learning at the digital frontier: a review of digital literacies in theory and practice. Journal of Computer Assisted Learning, 28: 547&ndash;556. doi: 10.1111/j.1365-2729.2011.00474.x McGreal, R. (2012). The need for Open Educational Resources for Ubiquitous Learning. 8th IEEE International Workshop on Pervasive Learning, Life and Leisure 2012 (pp. 679&ndash;684). Lugano. Radical Learning Group, La Trobe University. (2012) http://www.radicallearninggroup.blogspot.com.au/ University of Southern Queensland. (2012) Orion: Open Education Resources and Open Education Practices in Higher Education, http://adfi.usq.edu.au/projects/orion/2012/09/17/about/. Wiley, D. &amp; Gurrell, S. (2009). A decade of development... Open Learning: The Journal of Open and Distance Learning, 24(1), 11-21. Wiley, D. (2012). The MOOC Misnomer. Open Content. Wiley, D. &amp; Thanos, K. 2013. Getting Started with OER? Lumen Can Help! Webinar 18 April 2013 http://meet60543915.adobeconnect.com/p4so0obhcf1/&gt; last accessed 31 May 2013. Williams, S. (2002). Free as in Freedom: Richard Stallman&#39;s Crusade for Free Software. O&#39;Reilly Media, Inc. == Appendix 1 == === La Trobe University Conference on open education === The Open Education Working Group came together in early 2013 to organise a one-day conference to mark Open Education Week. The group noted a lack of acknowledgement of the worldwide open education event at other institutions in the region. The working group was subsequently tasked by the PVC Future Learning, Claire Macken, to provide recommendations on how to engage in the open education arena, following the Radical Learning Group&rsquo;s recommendation in its report of November 2012. The open education conference was intended to be a platform to discuss a wide range of issues related to open education, and to draw the conversation away from recent industry and media focus on MOOCs (Massively Open Online Courses). The group intended to highlight the lack of open education activity in our region and to change this by creating greater awareness within La Trobe. External guests were invited from a range of organisations outside La Trobe, as well as an extended network of colleagues from within La Trobe. === Conference findings === The one-day conference explored a range of issues relating to open educational practices, including distinctions between free access and open access. A full program, as well as summaries and recordings of all sessions, can be found on the wiki. ==== Development of capacity ==== Currently there is a lack of support and recognition structures in place for staff to adopt and create OER. Lack of skills (and awareness) in using and developing open resources is a barrier to adoption of open educational practices within La Trobe. If open teaching practices are to be valued, this needs to be demonstrated through practical measures such as recognition of contribution to open teaching resources, rather than current structure of recognising research publications only. Conference discussions demonstrated a need to develop capacity (institutional and individual) to produce, source and use open educational resources in teaching and learning. The 'Who&rsquo;s Doing What' session of the conference discussed Wikimedia Medicine project as an example of a project that is positioned and resourced to support our professional development and capacity building. ==== Flexibility in IP policy and procedures ==== There is a need for La Trobe&rsquo;s IP policy and copyright procedures to be clear and more flexible. This issue was raised in nearly all sessions of the conference. There was a specific suggestion to use &lsquo;Copyleft&rsquo; as an organising principle and there was debate about individual vs. institutional ownership of intellectual property. A proposed IP Policy was tabled, which was drafted for the University of Canberra and endorsed by the NTEU and others. The conference discussions gave support for individual ownership of work produced while working at La Trobe, as is the case in institutions such as the University of New South Wales, University of Technology Sydney, University of Western Australia, University of Wollongong, James Cook University and Australian National University. The conference also noted that while the Australian Department of Finance and Deregulation in 2010 advised the adoption of Creative Commons Attribution licence, in practice other Creative Commons licences have been applied to departmental content. The discussion concluded that for policy to flow through into practice, adequate explanation of the reasons for adopting a particular default Creative Commons licence <ref>{{Cite web|url=https://www.icslegal.co/|title=Apply for a Sponsor Licence {{!}} Step-by-Step Guidance UK & US|website=Sponsor Management System|language=en|access-date=2024-12-12}}</ref>must be given. ==== Existing open education projects around La Trobe University ==== A range of open education activities already in operation at La Trobe was discussed at the conference, from the centrally sanctioned iTunes U activities to faculty-driven efforts to provide educational materials on open access websites such as Wikipedia and Wikiversity. Conference attendees expressed a desire to have flexibility in their adoption of open education practices and use of open education platforms; that their options not be limited to one method such as through iTunes U. Discussions centred around how methods of engagement might change across disciplines and thus singular, stringent policies and guidelines would not be appropriate. Conference attendees also discussed their individual use and experience of open educational resources, citing difficulties in finding re-usable materials. The conference supported the development of open education practices in higher education in Australia. By opening up education, higher education institutions can build and develop strong connections and involvement with communities. {{CourseCat}} r4h83lm7pkkytkmh25ax46gtus7ff2h 1 159165 2691668 1168340 2024-12-12T17:41:16Z 152.26.89.206 /* Programming */ new section 2691668 wikitext text/x-wiki == About This Course == The [http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework] defines skills by grade level, with this course's levels equivalent to the following primary and secondary grade levels: * Basic - 3rd and 4th grade skills * Intermediate - 5th and 6th grade skills * Advanced - 7th and 8th grade skills * Expert - 9th and 10th grade skills In designing a course that could be used by adults, labeling the sections with primary and secondary grade levels was inappropriate. A search of level labeling for these types of activities returns basic, intermediate, advanced and expert as appropriate terms, so those are the designations this course uses. Basic, intermediate, and advanced are consistent with the levels of difficulty introduced. Expert would be expert only in comparison to the other levels, but would not qualify as expert in the typical industry certification sense. Those courses will be developed later under different titles. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:37, 29 March 2014 (UTC) == Programming == I think programming should be introduced at an earlier stage like intermediate or advanced. There are plenty of great starter options online like block based coding similar to scratch. [[Special:Contributions/152.26.89.206|152.26.89.206]] ([[User talk:152.26.89.206|discuss]]) 17:41, 12 December 2024 (UTC) olxjcz4fb8c2huamoyb8chchwh8x6rb 1 169166 2691646 1296610 2024-12-12T15:55:48Z DerGeist4040 2994919 /* Many minor spelling issues, some gibberish */ new section 2691646 wikitext text/x-wiki comment removed from resource, placed by [https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1271490&oldid=1240958]. WARNING: Some of the content of this page is incorrect. For example, Romulus did not found Rome, he was a myth. Rome was also brought down because of social issues, invasions from Germanic tribes, among other causes. While I can't prove this page is -entirely- incorrect, I advise you to use this resource at your own risk. -- unsigned, [[Special:Contributions/206.255.62.188|206.255.62.188]]. == Macedonian or Greek? == An anonymous user changed the text, I reverted but edited the text to reflect the distinction between Greece and Macedonia. The user reverted and asserted the "Macedonian" claim more strongly.[https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1296425&oldid=1296404] The distinction between Greece and Macedonia appears to be strong and important to some, however, the IP editor is apparently exaggerating as to the history of the phalanx. From [[w:Phalanx]] and [[w:Macedonian phalanx]] and [[w:Ancient Macedonian army]], the phalanx was developed by the Greek city-states and further developed by the Macedonians, their neighbors. I reverted. If the user wants to develop resources distinguishing between Greek and Macedonian culture and technology, that would be fine, if done with overall neutrality, I would assist on request or necessity. I'd recommend registering an account! --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 13:58, 5 March 2015 (UTC) == Many minor spelling issues, some gibberish == "The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers.” This section in particular sounds like complete gibberish, I tried to make as many qualities of life changes as I could, but I don't understand this section. Someone please try to fix it, as I cannot. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 15:55, 12 December 2024 (UTC) 654ug9jlcwh0bbvfrpef2gdhxt8c50g 2691652 2691646 2024-12-12T16:39:03Z DerGeist4040 2994919 /* Many minor spelling issues, some gibberish */ Reply 2691652 wikitext text/x-wiki comment removed from resource, placed by [https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1271490&oldid=1240958]. WARNING: Some of the content of this page is incorrect. For example, Romulus did not found Rome, he was a myth. Rome was also brought down because of social issues, invasions from Germanic tribes, among other causes. While I can't prove this page is -entirely- incorrect, I advise you to use this resource at your own risk. -- unsigned, [[Special:Contributions/206.255.62.188|206.255.62.188]]. == Macedonian or Greek? == An anonymous user changed the text, I reverted but edited the text to reflect the distinction between Greece and Macedonia. The user reverted and asserted the "Macedonian" claim more strongly.[https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1296425&oldid=1296404] The distinction between Greece and Macedonia appears to be strong and important to some, however, the IP editor is apparently exaggerating as to the history of the phalanx. From [[w:Phalanx]] and [[w:Macedonian phalanx]] and [[w:Ancient Macedonian army]], the phalanx was developed by the Greek city-states and further developed by the Macedonians, their neighbors. I reverted. If the user wants to develop resources distinguishing between Greek and Macedonian culture and technology, that would be fine, if done with overall neutrality, I would assist on request or necessity. I'd recommend registering an account! --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 13:58, 5 March 2015 (UTC) == Many minor spelling issues, some gibberish == "The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers.” This section in particular sounds like complete gibberish, I tried to make as many qualities of life changes as I could, but I don't understand this section. Someone please try to fix it, as I cannot. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 15:55, 12 December 2024 (UTC) :I may just remove this part, it looks like it doesn't add anything constructive to the article [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 16:39, 12 December 2024 (UTC) sfvb6a3teu55fh4ovo9vylga2kpj6i1 2691661 2691652 2024-12-12T16:54:45Z DerGeist4040 2994919 /* Still widely unfinished */ new section 2691661 wikitext text/x-wiki comment removed from resource, placed by [https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1271490&oldid=1240958]. WARNING: Some of the content of this page is incorrect. For example, Romulus did not found Rome, he was a myth. Rome was also brought down because of social issues, invasions from Germanic tribes, among other causes. While I can't prove this page is -entirely- incorrect, I advise you to use this resource at your own risk. -- unsigned, [[Special:Contributions/206.255.62.188|206.255.62.188]]. == Macedonian or Greek? == An anonymous user changed the text, I reverted but edited the text to reflect the distinction between Greece and Macedonia. The user reverted and asserted the "Macedonian" claim more strongly.[https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1296425&oldid=1296404] The distinction between Greece and Macedonia appears to be strong and important to some, however, the IP editor is apparently exaggerating as to the history of the phalanx. From [[w:Phalanx]] and [[w:Macedonian phalanx]] and [[w:Ancient Macedonian army]], the phalanx was developed by the Greek city-states and further developed by the Macedonians, their neighbors. I reverted. If the user wants to develop resources distinguishing between Greek and Macedonian culture and technology, that would be fine, if done with overall neutrality, I would assist on request or necessity. I'd recommend registering an account! --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 13:58, 5 March 2015 (UTC) == Many minor spelling issues, some gibberish == "The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers.” This section in particular sounds like complete gibberish, I tried to make as many qualities of life changes as I could, but I don't understand this section. Someone please try to fix it, as I cannot. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 15:55, 12 December 2024 (UTC) :I may just remove this part, it looks like it doesn't add anything constructive to the article [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 16:39, 12 December 2024 (UTC) == Still widely unfinished == This article needs many more changes, do not be scared to revise any of my editing or to change the article. If you feel like any major changes are needed, post it here and ask. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 16:54, 12 December 2024 (UTC) 3mg5fztqs8y8u7exldtfmiglef4kvx7 2691677 2691661 2024-12-12T18:21:46Z Atcovi 276019 /* Many minor spelling issues, some gibberish */ Reply 2691677 wikitext text/x-wiki comment removed from resource, placed by [https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1271490&oldid=1240958]. WARNING: Some of the content of this page is incorrect. For example, Romulus did not found Rome, he was a myth. Rome was also brought down because of social issues, invasions from Germanic tribes, among other causes. While I can't prove this page is -entirely- incorrect, I advise you to use this resource at your own risk. -- unsigned, [[Special:Contributions/206.255.62.188|206.255.62.188]]. == Macedonian or Greek? == An anonymous user changed the text, I reverted but edited the text to reflect the distinction between Greece and Macedonia. The user reverted and asserted the "Macedonian" claim more strongly.[https://en.wikiversity.org/w/index.php?title=Greek_and_Roman_military_traditions&diff=1296425&oldid=1296404] The distinction between Greece and Macedonia appears to be strong and important to some, however, the IP editor is apparently exaggerating as to the history of the phalanx. From [[w:Phalanx]] and [[w:Macedonian phalanx]] and [[w:Ancient Macedonian army]], the phalanx was developed by the Greek city-states and further developed by the Macedonians, their neighbors. I reverted. If the user wants to develop resources distinguishing between Greek and Macedonian culture and technology, that would be fine, if done with overall neutrality, I would assist on request or necessity. I'd recommend registering an account! --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 13:58, 5 March 2015 (UTC) == Many minor spelling issues, some gibberish == "The soldier then drew his sword and put his shield in front of him the Gauls attack in the last moment the Romans in the first line strikes the enemy with his shield he then raises his shield goes under the shield and stabs the enemy in the stomach he then goes to the side the soldiers in the 2 row steps in the enemy retreats then soldiers in the first row rotates and places themselves in the last line this meant that the enemy would always fight rested soldiers.” This section in particular sounds like complete gibberish, I tried to make as many qualities of life changes as I could, but I don't understand this section. Someone please try to fix it, as I cannot. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 15:55, 12 December 2024 (UTC) :I may just remove this part, it looks like it doesn't add anything constructive to the article [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 16:39, 12 December 2024 (UTC) ::Hey {{ping|DerGeist4040}} you're free to make any changes you wish, as long as they subscribe to our [[Wikiversity:Mission|mission page]]! Go ahead and make the changes you deem are necessary to improve the page's quality. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:21, 12 December 2024 (UTC) == Still widely unfinished == This article needs many more changes, do not be scared to revise any of my editing or to change the article. If you feel like any major changes are needed, post it here and ask. [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 16:54, 12 December 2024 (UTC) tro9ik8wr2pcklxbkrxk7m6ndvx03en 0 171005 2691624 2691407 2024-12-12T13:43:05Z Young1lim 21186 /* Geometric Series Examples */ 2691624 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241212.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] j3zm513o8jqa5o2xkzvuhyl2ybqub3v 1 198621 2691722 1641663 2024-12-12T23:47:45Z Huz and Buz 2928717 /* Joshua */ new section 2691722 wikitext text/x-wiki Just wondering which author/book has been used for the source divisions in this project? Does anyone know? [[User:Arswann|Arswann]] ([[User talk:Arswann|discuss]] • [[Special:Contributions/Arswann|contribs]]) 18:55, 11 July 2015 (UTC) :I asked the same question. I got no response.--[[User:Jcvamp|Jcvamp]] ([[User talk:Jcvamp|discuss]] • [[Special:Contributions/Jcvamp|contribs]]) 18:44, 15 December 2016 (UTC) == Joshua == @[[User:Lucasrbrown23|<bdi>Lucasrbrown23</bdi>]] What's the point of adding Joshua? It's not typically included in the Documentary Hypothesis. [[User:Huz and Buz|Huz and Buz]] ([[User talk:Huz and Buz|discuss]] • [[Special:Contributions/Huz and Buz|contribs]]) 23:47, 12 December 2024 (UTC) mpa9mbqjft5yddjh3xwckryicsrs40l 0 203942 2691714 2691530 2024-12-12T23:09:18Z Young1lim 21186 /* Lambda Calculus */ 2691714 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20241213.pdf |pdf]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] qiqt790rfjfl7lh0891g66ppowruvnh 3 206779 2691673 2660995 2024-12-12T18:16:41Z Dan Polansky 33469 2691673 wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Bert Niehaus!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:Insert-signature.png]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. * Follow Wikiversity on [[twitter]] (http://twitter.com/Wikiversity) and [[identi.ca]] (http://identi.ca/group/wikiversity). </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:27, 23 January 2016 (UTC)</div> {{Robelbox/close}} == Subpages and Links == Learning projects are best organized with [[Wikiversity:Subpages|subpages]] (like Wikibooks) rather than as individual pages (like Wikipedia). I've moved the E-Proof Future Development page to [[E-Proof/Future Development and Feature Discussion]]. When linking to Wikiversity pages, use internal links with <nowiki>[[Title]]</nowiki> or <nowiki>[[Title|Display]]</nowiki> syntax. When linking to subpages, use <nowiki>[[/Subpage/]]</nowiki> syntax. See [[Making links]] for more information. Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:26, 20 December 2016 (UTC) Thank you Dave for your support == Risk Management == Hi Bert Niehaus! Your resource [[Risk Management]] appears to be ready for learners! Would you like it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 13:52, 1 May 2017 (UTC) Thank you for announcement offer, Excuse me for late reply. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:45, 14 August 2017 (UTC) == Trust == I see you have recently added both [[Trust in Data, Information and Knowledge]] and [[Trust in Capacity Building Material]]. Are you ultimately building a learning project on [[Trust]] that these and any additional pages belong to, or are they part of [[Risk Management]]? We try to organize Wikiversity content by learning project rather than just learning resource. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:26, 11 July 2017 (UTC) (1) First I though I just plan a trust learning resource for risk management, but then I looked on my lecture about encryption, digital signature and number theory it makes sense to me to broaden the mathematical perspective on encryption to application and trust. On the other hand from trust it makes sense to learn about digital signature and encryption. So it makes more sense to build a learning resource on trust. Would be great to have some psychologist on board to approach the trust topic from their angle, but who knows collaborative development of content could trigger wonderful things, Thank you for your support, Dave [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 16:20, 12 July 2017 (UTC) == Wikipedia Imports == All Wikimedia wikis are licensed using Creative Commons CC-BY-SA licensing. The BY part of that license requires crediting the source whenever licensed content is reused. Within wiki software, the best way to credit the source is by including their edits in the page history. This is done using [[Special:Import]], available to Wikiversity custodians and curators. If you would like to import any additional Wikipedia pages, please either use [[Wikiversity:Import]] to request page imports or see [[Wikiversity:Curators]] to request curatorship so you may process your own imports. Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 12 August 2017 (UTC) [[Special:Import]] shows a 'permission error' for me as an ordinary wiki author --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:40, 13 August 2017 (UTC) :Is the curator status the only option as a user to do branching of content development from Wikipedia to Wikiversity? ::Yes, the only way to import pages from Wikipedia yourself is with curator status. Otherwise, you are welcome to post at [[Wikiversity:Import]] and one of us can import pages for you. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:48, 12 August 2017 (UTC) == Forking and Tailor Eductional Resources == * '''[[/Forking/|Forking of Learning Resources and tailor Learning Resources for Target Groups]]''' * '''[[Open Educational Resources/Localization]]''' === Teachers Informations: Forking Info for Learner Profile === Sometime a learning topic can be developed for different educational level. Teacher information can be used to fork from a parent module to submodules tailored for the target group. That is my solution to support branching within the existing IT infrastructure. See '''[[Water#Information_for_Teachers|Water]]''' learning resource. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:45, 29 September 2017 (UTC) == Curator Status == Based on the support expressed at [[Wikiversity:Candidates for Custodianship/Bert Niehaus]], you are now a curator. Congratulations! You should see new tools available under [[Special:SpecialPages]] and on the More menu on each page. Use them wisely, and let me know if you have any questions. Also, please update [[Wikiversity:Support staff]] and add your information. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:48, 22 August 2017 (UTC) Ttank you very much, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 04:54, 23 August 2017 (UTC) == Sustainable Development Goals == Hi Bert Niehaus! Your resource [[Sustainable Development Goals]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:47, 29 August 2017 (UTC) OK thank you, learning tasks are still missing. And I want to insert the SDG logos on pages, but have to read licensing by United Nations before using them in Wikiversity. SDG Logos seem not possible to import from Wikipedia. :* I've looked at the logos at the Wikipedia [[w:Sustainable Development Goals|Sustainable Development Goals]]. One is at Commons and the other is fair use which is also allowed here by USA copyright law. You are free to upload them here or use the Commons version, where available. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 14:08, 29 August 2017 (UTC) :* Thank you --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 08:49, 30 August 2017 (UTC) Logos inserted and started to assign Sustainable Developement Goals to Learning Resources e.g. on [[Water]] and [[Collaborative Mapping]] == PanDocElectron and Water == Hi Bert Niehaus! Your resources [[PanDocElectron]] and [[Water]] appear to be ready for learners! Would you like to have them announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 12:12, 24 October 2017 (UTC) Thank you for regarding the learning resource as ready. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 13:46, 24 October 2017 (UTC) == Commercial Data Harvesting == Hi Bert! I love your page on [[Commercial Data Harvesting]] and its 5 basic constituents. I'm currently writing an article on the subject and was wondering if there's any literature that mentions the concept (and its constituents) like that Thank you, Ana Leonardi [[User:Leonardiac|Leonardiac]] ([[User talk:Leonardiac|discuss]] • [[Special:Contributions/Leonardiac|contribs]]) 15:06, 29 October 2017 (UTC) == 3D Modelling == Hi Bert Niehaus! Your learning resource [[3D Modelling]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:54, 21 January 2018 (UTC) Yes, fine, thank you for annoucing it. Still working on the 3D modelling, but some learners might benefit from the resource. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 16:57, 21 January 2018 (UTC) == Expert Focus Group for Space and Global Health == Hi Bert Niehaus! Your resource [[Expert Focus Group for Space and Global Health]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:30, 5 March 2018 (UTC) There is lot of work to do, but yes if you think it is worth publishing, it is Ok for me. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:59, 6 March 2018 (UTC) == Share your experience and feedback as a Wikimedian in this global survey == <div class="mw-parser-output"> <div class="plainlinks mw-content-ltr" lang="en" dir="ltr"> Hello! The Wikimedia Foundation is asking for your feedback in a survey. We want to know how well we are supporting your work on and off wiki, and how we can change or improve things in the future. The opinions you share will directly affect the current and future work of the Wikimedia Foundation. 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We have designed the survey to make it impossible to identify which users have taken the survey, so we have to send reminders to everyone. To opt-out of future surveys, send an email through EmailUser feature to [[:m:Special:EmailUser/WMF Surveys|WMF Surveys]]. You can also send any questions you have to this user email. [[m:Community_Engagement_Insights/About_CE_Insights|Learn more about this survey on the project page.]] This survey is hosted by a third-party service and governed by this Wikimedia Foundation [[:foundation:Community_Engagement_Insights_2018_Survey_Privacy_Statement|privacy statement]]. </div> <span class="mw-content-ltr" dir="ltr">[[m:User:WMF Surveys|WMF Surveys]]</span>, 00:44, 20 April 2018 (UTC) </div> <!-- Message sent by User:WMF Surveys@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Community_Engagement_Insights/MassMessages/Lists/2018/ot6&oldid=17952438 --> == Think globally - act locally == Hi Bert Niehaus! Your resource [[Think globally - act locally]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 03:00, 23 April 2018 (UTC) Thank you Marshall, It is ok, publishing the resource. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 05:09, 28 April 2018 (UTC) == Structuring Data == Hi Bert Niehaus! Your resource [[Structuring Data]] appears well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 03:39, 24 July 2018 (UTC) Thank you Marshall, I am Ok with publishing the resource --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 07:38, 25 July 2018 (UTC) == Humanitarian Open Streetmap == Hi Bert Niehaus! Your resource [[Humanitarian Open Streetmap]] appears to be well-developed and ready for learners and participants! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:39, 28 August 2018 (UTC) Thank you Marshallsumter, I am Ok with publishing the resource --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:20, 2 September 2018 (UTC) == Collaborative mapping == Hi Bert Niehaus! Your resource [[Collaborative mapping]] appears well-developed and ready for learners and participants! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:35, 24 September 2018 (UTC) Thank you Marshall, I am OK with annoucement, --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:59, 26 November 2018 (UTC) == Moving Average == Hi Bert Niehaus! Your statistics resource [[Moving Average]] is well-developed and appears ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:40, 10 December 2018 (UTC) Dear Marshall, currently Moving Average needs some learning tasks to be added. Sorry for late reply. Best regards, bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 18:01, 21 February 2019 (UTC) :: Dear [[User:Marshallsumter|Marshallsumter]] the learning resource Moving Average can be annouced on Main Page News - == Systems Thinking == Hi Bert Niehaus! Your landing page [[Systems Thinking]] appears well-developed and ready to assist learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:13, 18 February 2019 (UTC) Dear Marshall, I need a bit of work for systems thinking to be ready. I would recommend that is NOT announced on the Main page. Will come back to you, according to this learning module. Thank you, Bert --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 18:00, 21 February 2019 (UTC) == Community Insights Survey == <div class="plainlinks mw-content-ltr" lang="en" dir="ltr"> '''Share your experience in this survey''' Hi {{PAGENAME}}, The Wikimedia Foundation is asking for your feedback in a survey about your experience with {{SITENAME}} and Wikimedia. The purpose of this survey is to learn how well the Foundation is supporting your work on wiki and how we can change or improve things in the future. The opinions you share will directly affect the current and future work of the Wikimedia Foundation. Please take 15 to 25 minutes to '''[https://wikimedia.qualtrics.com/jfe/form/SV_0pSrrkJAKVRXPpj?Target=CI2019List(other,act5) give your feedback through this survey]'''. It is available in various languages. This survey is hosted by a third-party and [https://foundation.wikimedia.org/wiki/Community_Insights_2019_Survey_Privacy_Statement governed by this privacy statement] (in English). Find [[m:Community Insights/Frequent questions|more information about this project]]. [mailto:surveys@wikimedia.org Email us] if you have any questions, or if you don't want to receive future messages about taking this survey. Sincerely, </div> [[User:RMaung (WMF)|RMaung (WMF)]] 14:34, 9 September 2019 (UTC) <!-- Message sent by User:RMaung (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=CI2019List(other,act5)&oldid=19352874 --> == Reminder: Community Insights Survey == <div class="plainlinks mw-content-ltr" lang="en" dir="ltr"> '''Share your experience in this survey''' Hi {{PAGENAME}}, A couple of weeks ago, we invited you to take the Community Insights Survey. It is the Wikimedia Foundation’s annual survey of our global communities. We want to learn how well we support your work on wiki. We are 10% towards our goal for participation. If you have not already taken the survey, you can help us reach our goal! '''Your voice matters to us.''' Please take 15 to 25 minutes to '''[https://wikimedia.qualtrics.com/jfe/form/SV_0pSrrkJAKVRXPpj?Target=CI2019List(other,act5) give your feedback through this survey]'''. It is available in various languages. This survey is hosted by a third-party and [https://foundation.wikimedia.org/wiki/Community_Insights_2019_Survey_Privacy_Statement governed by this privacy statement] (in English). Find [[m:Community Insights/Frequent questions|more information about this project]]. [mailto:surveys@wikimedia.org Email us] if you have any questions, or if you don't want to receive future messages about taking this survey. Sincerely, </div> [[User:RMaung (WMF)|RMaung (WMF)]] 19:13, 20 September 2019 (UTC) <!-- Message sent by User:RMaung (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=CI2019List(other,act5)&oldid=19395141 --> == Reminder: Community Insights Survey == <div class="plainlinks mw-content-ltr" lang="en" dir="ltr"> '''Share your experience in this survey''' Hi {{PAGENAME}}, There are only a few weeks left to take the Community Insights Survey! We are 30% towards our goal for participation. If you have not already taken the survey, you can help us reach our goal! With this poll, the Wikimedia Foundation gathers feedback on how well we support your work on wiki. It only takes 15-25 minutes to complete, and it has a direct impact on the support we provide. Please take 15 to 25 minutes to '''[https://wikimedia.qualtrics.com/jfe/form/SV_0pSrrkJAKVRXPpj?Target=CI2019List(other,act5) give your feedback through this survey]'''. It is available in various languages. This survey is hosted by a third-party and [https://foundation.wikimedia.org/wiki/Community_Insights_2019_Survey_Privacy_Statement governed by this privacy statement] (in English). Find [[m:Community Insights/Frequent questions|more information about this project]]. [mailto:surveys@wikimedia.org Email us] if you have any questions, or if you don't want to receive future messages about taking this survey. Sincerely, </div> [[User:RMaung (WMF)|RMaung (WMF)]] 17:04, 4 October 2019 (UTC) <!-- Message sent by User:RMaung (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=CI2019List(other,act5)&oldid=19435548 --> ==A barnstar for you== {| style="border: 1px solid {{{border|gray}}}; background-color: {{{color|#fdffe7}}};" |rowspan="2" style="vertical-align:middle;" | {{SAFESUBST:<noinclude />#ifeq:{{{2}}}|alt|[[File:Kindness Barnstar Hires.png|100px]]|[[File:Random Acts of Kindness Barnstar.png|100px]]}} |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Random Acts of Kindness Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | {{{Thank you for supporting my proposal on WikiQuiz}}} |} [[User:RIT RAJARSHI|RIT RAJARSHI]] ([[User talk:RIT RAJARSHI|discuss]] • [[Special:Contributions/RIT RAJARSHI|contribs]]) 16:27, 2 April 2020 (UTC) == SIR model and COVID == Thank you for the suggestion on my page ([[COVID-19/Iluvalar]]) , I think you misunderstood the object of that chapter completely. The point is to explain that the first exponential part of the data, can only be the test production curve and have nothing to do with the actual virus. The starting data and the production rate of the virus test had it own value and any contact with the real infection curve of the virus could only be accidental and highly unlikely. For the actual SIR model I used, see the line "model apr 4" and "model apr 7" later on the page, I'll try to add the data for the following month, but I think I was pretty accurate with that. [[User:Iluvalar|Iluvalar]] ([[User talk:Iluvalar|discuss]] • [[Special:Contributions/Iluvalar|contribs]]) 01:45, 10 May 2020 (UTC) : {{At|Iluvalar}} Agree with you, testing is not performed in a randomly selected cohort of citizens and testing is not driven by a controlled study design. Asymptomatic patients are not tested and even symptomatic patients were not tested because staff in health care facilities did not decide to test the patient or decided to test other patients with higher priority first, because of the limited test capacity. Shall we add a specific submodul to specify the limitation of modelling due to the limitation of dada aquisitions. Or do you have other suggestions for learning resource? --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:26, 11 May 2020 (UTC) == Your class on Commercial Data Harvesting == Hi {{PAGENAME}}: Since I discovered it I have been following your class on [[Commercial Data Harvesting]] with interest. I am fairly new at WV and still have not figured out what users like me who are not lecturers at WV are allowed to do. At Universities that attended in person as a student, free exchange of information was tolerated if not encouraged. Anyway, what I came here to suggest is that the course assumption that '''...users regard themselves as ''customer'' of an provider of a free digital service''', may have been accurate in the past at some [[w:Big tech]] offerings, in some countries, but it is more questionable if all users of such services have always considered themselves as a customer of a free service. : Thank you for your feedback, added a comment according to your feedback for the learning resoruce --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 11:56, 29 November 2020 (UTC) Take for example the user revolt reffered to in [[w:Reddit#Company_history]], hardly the action of individuals gratefull for a free service they are receiving? I hope I am making sense, contributing positively to WV, and not wasting your time unnecessarily. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 19:30, 28 November 2020 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:Data analysis digital learning environments.svg]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 00:25, 9 January 2022 (UTC) :: DONE --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:48, 8 February 2022 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:Eye image smoother.png]] * [[:File:Eye image.png]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 19:45, 7 February 2022 (UTC) :: DONE --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:48, 8 February 2022 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:Cas4wiki settings create url.png]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 05:15, 23 December 2022 (UTC) == Help debugging my image map == I have attempted to create an image map at: [[The Wise Path]] I followed the (Wikipedia) image map example. The map does not work. I will appreciate your help in debugging my image map. Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:49, 23 May 2023 (UTC) :Excuse me for the late reply [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 10:04, 25 May 2023 (UTC) : I fixed it! thanks == Lightboard == FYI, I marked [[Lightboard]] for proposed deletion since it is empty. Perhaps you can add a sentence or two and a couple of good links to make the page worthwhile? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:01, 3 October 2024 (UTC) == Complex Analysis/Curves == In [[Complex Analysis/Curves]], I find edits by [[User:Eshaa2024]] for which I cannot quickly confirm are good. You are the author of the page, so you may want to have a look. Perhaps Eshaa2024 is your collaborator; I don't know. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:16, 12 December 2024 (UTC) mvk8aasvdo12ohrfpjyoy5fgp6dl63t 0 207226 2691773 2687752 2024-12-13T09:29:22Z Elominius 2911372 /* Examples */ + 2 functions 2691773 wikitext text/x-wiki Functions allows you to group pieces of code and reuse it along your programs. <ref>http://tldp.org/HOWTO/Bash-Prog-Intro-HOWTO-8.html</ref> == Pre-requisites == Now is the time to note, while some concepts are assumed as a pre-requisite, it's reasonable to expect not every student will be familiar with all concepts. Therefore, where a concept is listed here in the pre-requisites, it will be introduced with a link to a more detailed explanation. '''Environment:''' * have completed the course Introduction * use ENVIRONMENT variables * stdin, stdout * re-direct outputs '''Commands:''' * how to define a function * '''expr''' -- does arithmetic, * '''tail''' -- delivers last output lines * <code>[[history]]</code> -- delivers command history == New Concepts == '''commands:''' * declare -- to view functions * history -- to view command history * set -- positional parameters * source -- to re-use, or re-load functions '''shell features''' * positional parameters * environment variables * sub-shell execution == Use function arguments == In this exercise, we will write two functions which use command arguments. The first function will display a function body or bodies. The second will display our command history <ref>https://en.wikipedia.org/wiki/History_(Unix)</ref>.. It will have an optional single argument, a number, which will default to the number of LINES, an [[Wikipedia:Environment variable|Environment variable]], in your terminal window. In the course of the exercise we will: * use a single function argument * usa a variable number of arguments * use a default argument * use a function to display our function repertoire. '''View a function body or bodies:''' First, from our last section, recall: $ helloWorld () { echo "Hello, World!"; } $ declare -f helloWorld ... shows the function body, Does your output meet your expectations. If this is the first time you've tried it, you will see something different than you entered. Let's write a function to do this: $ fbdy () { declare -f $*; } $ fbdy helloWorld fbdy Now you are less surprised by the output. What's the '''$*''' doing? It says "when this command (or function) is executed, put all the blank-separated positional parameters right here". You observe the function is little different from using the built-in command declare with it's -f option. Like an alias, but the function allows upward-compatible evolution. In this case, a version of '''fbdy''' may return one-line functions as one-liners, another may routinely add (or remove) function tracing to the function body. Make sure you've executed the examples above. == Use command history == '''How many lines is our terminal displaying?''' An Environment variable, LINES is often set by the shell to the number of lines the terminal window is displaying. $ echo $LINES If you see nothing, then variable is not set on your terminal. We'll supply a default. It is useful to define a function, again almost an alias '''th''', standing for '''T'''ail '''H'''istory. The reason for this function goes a bit beyond our needs here. However it's useful as a short-hand, if not not to provide a consistent interface to different shell's with a different sets of options for a single command. $ set $(expr ${LINES:-27} - 3); history | tail -$1 execute that command; if needed, look ahead to subshells here is a sample of results: $ set $(expr ${LINES:-27} - 3); history | tail -$1 683 pushd $(which cmdlib) 684 pushd $(which $(dirname cmdlib)) 685 pushd 686 pushd $(dirname $(which cmdlib)) 687 view cmdlib 688 ff th 689 history -24 690 th 691 th 692 ff th 693 set | grep HISTSIZE 694 set | grep HIST 695 th () { set $1 $(expr ${LINES:-27} - 3); history | tail -$1; } 696 declare -f th 697 echo $LINES 698 clear 699 declare -f th 700 th 24 701 th 702 echo $LINES 703 history 24 704 history 45 705 history -h 706 history -T 707 history -x 708 uname -a 709 https://www.gnu.org/software/bash/manual/html_node/Bash-History-Builtins.html#index-history-builtins 710 th () { set $1 $(expr ${LINES:-27} - 3; history | tail -$1; } 711 th () { set $1 $(expr ${LINES:-27} - 3); history | tail -$1; } 712 echo $LINES 713 set $(expr ${LINES:-27} - 3); history | tail -$1 bin.$ Notice a number of things about that command: * number 713 is the last command itself * thirty-one commands were displayed Now, convert that into a function, and test it. $ th () { set $1 $(expr ${LINES:-27} - 3); history | tail -$1; } $ th # returns your recent command history, filling up the screen It turns out in the bash shell, history is a [[Wikipedia:Shell builtin|builtin]], which shows a complete list of [https://www.gnu.org/software/bash/manual/html_node/Bash-History-Builtins.html bash history builtins here]. What happened here? The [https://www.mkssoftware.com/docs/man1/set.1.asp set command (ksh version)] in this usage [https://gnu.org/software/bash/manual/html_node/Shell-Parameter-Expansion.html assigns the positional parameters]. If the function is used with one, then its done like this: $ th 24 # 24 -> $1 ... so, the command becomes "history | tail -24" If no argument is used, then it becomes $ th # and if no lines are set, then: expr 27 - 3 ( = 24 ) , so ... "history | tail -24" again but if LINES was say, 34, then it's [http://faculty.salina.k-state.edu/tim/unix_sg/bash/math.html expr] 34 - 3 (= 31) ... history | tail -31 The shell parameter substitution works for named variables (LINES, SHELL, ...) as well as the [https://gnu.org/software/bash/manual/html_node/Shell-Parameter-Expansion.html positional parameters] (1, 2, ... *) and this expression is most useful to [http://wiki.bash-hackers.org/syntax/pe#use_a_default_value assign a default value], in this example: $ echo ${LINES:-27} And this last feature introduced here is the ability to insert sub-shell results in the command. The general idea is: $ command .. $( sub-shell command or commands... ) ... where the results of ''sub-shell command or commands...'' is inserted into the command ... In our case then, the result of the shell arithmetic is inserted: $ set "" $(expr $LINES - 3) # $1 was empty, becomes $ set $(expr 34 - 3) # to be evaluated, $ set 31; history | tail -$1 # then becomes $ history | tail -34 == Display functions == To collect our new and useful work, lets' see what we have: $ fbdy fbdy th For example: bin.$ fbdy fbdy th fbdy () { declare -f $* } th () { set $1 $(expr ${LINES:-27} - 3); history | tail -$1 } bin.$ This is progress. Do your results compare? == Edit functions == == Save functions == You have at least two ways to keep a consistent set of functions available for your command line: * save them in a local file * save them to load when you login We'll exercise both methods here. First the local file. You'll find that not all functions are needed in all instances. === In a local file === Functions can be stored in any <code>.sh</code> file. To load the functions from that file, run the command <code>. filename.sh</code>. === load at login === Functions can be stored in the file named <code>.bashrc</code>. It is typically located in the home directory (shortcut: <code>~/.bashrc</code>). == Reload functions == == Examples == * Create a directory and enter it: ** <code>mkcd() { if [ ! -d "$@" ];then mkdir -p "$@" ;fi; cd "$@"; }</code> * Change the window title in a terminal emulator: ** <code>window_title() { printf "\033]0;$*\007"; }</code> * Prevent a hard drive that does not respond to <code>hdparm</code> from spinning down: ** <code>spindisk() { while : ; do (sudo dd if=/dev/$1 of=/dev/null iflag=direct ibs=4096 count=1; sleep 29); done }</code> ** Requires root access to run due to block-level disk access. ** Adjust the time after <code>sleep</code> to just below your hard drive's default spin-down timeout. ** Exit with <kbd>CTRL</kbd>+<kbd>C</kbd>. * Count the files in a directory: <code>filecount() { ls "$@" |wc -l; }</code> * Count the files in a directory and its subdirectories: <code>filecount-total() { find "$@" |wc -l; }</code> * Count the folders only: <code>subfoldercount() { find "$@" -type d |wc -l; }</code> * Search for files in a directory: <code>findfile() { find "$2" |grep -i "$1"; }</code> * Search text inside 7z (7-Zip) archives: <code>7zgrep() { 7z e -so "$2" |grep -i "$1"; }</code> * Find the newest or the oldest file in a given directory: ** <code>newestfile() { find "$@" -type f -printf '%T+ %p\n' |sort |tail -n 1; }</code> ** <code>oldestfile() { find "$@" -type f -printf '%T+ %p\n' |sort |head -n 1; }</code> * Add time stamps at the beginning of specified file names: <syntaxhighlight lang=sh> prepend_timestamps() { if [[ "$@" == "" ]]; then echo "No file name specified. Exiting."; return 1; fi; for filename in "$@"; do timestamp="$(date -r "$filename" +%Y-%m-%dT%H-%M-%S)" mv -nv -- "$filename" "$timestamp $filename"; done; } alias addtimestamps=prepend_timestamps; </syntaxhighlight> * Generate check sums of all files in the current or a specified directory: <syntaxhighlight lang=sh> superMD5() { find "$@" -type f |sort |xargs -d '\n' md5sum; } </syntaxhighlight> === Working with multimedia === * Create a file list for the concatenation feature of <code>ffmpeg</code> from the <code>find</code> command: ** <code>fffile() { sed -r "s/(.*)/file '\1'/g"; }</code> ** Example use with pipe: <code>find DCIM/Camera/VID_{{#time:Ymd}}*.mp4 |fffile >>example.txt </code> * Access the concatenation feature of [[:w:ffmpeg|ffmpeg]] with only two parameters: ** <code>ffconcat() { ffmpeg -f concat -safe 0 -i "$1" -c copy "$2"; }</code> ** Example use: <code>ffconcat example.txt example.mp4</code> * Verify the integrity of multimedia files using its decoding mechanism, independently from the file system: <code>fferror() { ffmpeg -v error -i "$1" -f null - ;}</code> * Generate a table of video resolutions and framerates by processing the text generated by the [[:w:mediainfo|mediainfo]] tool. ** <code>mediainfotable() { mediainfo "$@" |grep -v "Frame rate.*SPF" |grep -P "(name|Width|Height|Frame rate )" |tr '\n' ' ' |sed -r 's/FPS/FPS\n/g' |sed -r "s/( )+//g"; }</code> * Redact geolocation from video files before sharing for privacy. This will overwrite the specified video files in-place, so it is recommended to only use it on copies of video files. ** <code>gpsnull() { sed -i -r "s/\+[0-9][0-9]\.[0-9][0-9][0-9][0-9]\+[0-9][0-9][0-9]\.[0-9][0-9][0-9][0-9]\//+00.0000+000.0000\//g" "$@"; }</code> * Mute the audio of a video (specify input and output file): <code>ffmute() { ffmpeg -i "$1" -c:v copy -an "$2"; }</code> * Extract the audio from a video file intp a separate file (specify input and output file): <code>ffaudio() { ffmpeg -i "$1" -c:a copy -vn "$2"; }</code> * Find out the total size of a selection of files: <code>totalsize() { du -sh -c "$@" |tail -n 1; }</code> * Get the frame rate of a video: <code>getfps() { ffprobe -v 0 -of csv=p=0 -select_streams v:0 -show_entries stream=r_frame_rate "$1"; }</code> * Get the resolution of a video: <code>getRes() { echo $(ffprobe -v 0 -of csv=p=0 -select_streams v:0 -show_entries stream=height "$1"; )p; }</code> * Stick all videos from one folder into one video file without re-encoding (only works for files from the same device with the same width and height and frame rate): <syntaxhighlight lang=sh> ffconcat-dir() { # edge case handlers if [[ "$1" == "" ]]; then echo "No directory specified. Exiting." ; return 1; fi if [[ "$1" == "ffconcat-dir.txt" ]]; then echo "This name is reserved. Please choose a different directory."; return 1; fi if [ -d "ffconcat-dir.txt" ]; then mv ffconcat-dir.txt "ffconcat-dir (usurped-$(date +%Y%m%d%H%M%S))"; fi # main part truncate -s 0 ffconcat-dir.txt # blanking temporary file from last run find "$1" -maxdepth 1 -type f |sed -r "s/(.*)/file '\1'/g" >ffconcat-dir.txt if [[ "$2" != "" ]]; then outfile="$2"; else outfile=tmp.mp4;fi ffmpeg -f concat -safe 0 -i ffconcat-dir.txt -c copy "$outfile" } </syntaxhighlight> Example use: <code>ffconcat-dir folder_name output_video_name.mp4</code> This works for other file types too, but not with mixed file types. The extension specified file type has to match the type of the source files. The default name is <code>tmp.mp4</code> if none is specified. == References == {{subpage navbar}} [[Category:Bash programming]] 02i2voti3bd4geds9845fnfb5u0z2zf 0 208687 2691698 2682682 2024-12-12T22:20:20Z The Love Atom 2980440 Typo, distinguish instead of distinguishing 2691698 wikitext text/x-wiki —Thinking Together ==Introduction== [[File:Kathy Matayoshi and Mazie Hirono.jpg|thumb|Two people practicing dialogue]] {{TOC right | limit|limit=2}} We have suddenly gone beyond ordinary conversation and are now beginning to listen, truly understand, learn from each other, and create together as we communicate candidly. We are thinking together, meaning now flows freely, and we are learning from the transformation that is [[w:dialogue|dialogue]]. '''Objectives''' The objectives of this course are to: *Recognize various forms of communication. *Understand the benefits of using dialogue to communicate. *Learn to use dialogue as your preferred method of communication. *Experience a synthesis and interweaving of ideas. *Gain insight as you dialogue with others. {{100%done}}{{By|lbeaumont}} The course contains many [[w:Hyperlink|hyperlinks]] to further information. Use your judgment and these [[What Matters/link following guidelines|link following guidelines]] to decide when to follow a link, and when to skip over it. This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]]. This material has been adapted from the EmotionalCompetency.com [http://emotionalcompetency.com/dialogue.htm page on dialogue], with permission of the author. If you wish to contact the instructor, please [[Special:Emailuser/Lbeaumont | click here to send me an email]] or leave a comment or question on the [[Talk:Practicing_Dialogue|discussion page]]. [[File:Practicing Dialogue Audio Dialogue.wav|thumb|Practicing Dialogue Audio Dialogue]] ==Dialogue is Distinct== [[w:Dialogue|Dialogue]] is the creative thinking together that can emerge when genuine [[w:Empathy|empathetic]] [[Knowing Someone/Deep Listening|listening]], respect for all participants, safety, [[w:Peer_group|peer]] relationships, suspending [[w:judgment|judgment]], sincere inquiry, courageous speech, and discovering and disclosing assumptions work together to guide our conversations. It is an activity of curiosity, cooperation, creativity, discovery, and learning rather than persuasion, competition, fear, and conflict. Dialogue is the only [[w:Symmetry|symmetrical]] form of communication. Dialogue emerges from [[w:Trust_(social_sciences)|trusting]] relationships. Dialogue is a form of conversation that is distinct from [[w:Conversation#Discussion|discussion]], [[w:debate|debate]], distraction, dismissal, delegation, disingenuous, diatribe, and [[w:dogma|dogma]] because dialogue is the only form of communication where the participants act as authentic peers. All other [[Communicating Power|forms of communication]] emphasize a [[w:Power_social_and_political|power relationship]] that interferes with the synthesis, analysis, and interweaving of ideas that characterize dialogue. Dialogue is driven by genuine curiosity and [[w:respect|respect]] rather than by power. ''Deliberation'' describes a period of thought and reflection that can take place during any conversation. [[w:Rapport|Rapport]] is a close synonym to dialogue. ===Assignment=== *Listen to conversations and various other communications such as advertisement, advocacy, opinions, debates, etc. *As you are [[Knowing Someone/Deep Listening|listening]], identify the [[Communicating Power|form of communication]] according to the power relationships being displayed. Name the form of communications as being: dialogue, discussion, debate, defense, distraction, dismissal, delegation, disingenuous, dialectic, decree, diatribe, or dogma. * View the video [https://www.youtube.com/watch?v=0wnujWPW5U0 The solution is in the dialogue], by Peter Nixon presented April 2014 at TEDxHKUST *How often do you witness skillful dialogue? ==Toward Dialogue== The goal of dialogue is ''insight'', the goal of [[w:argumentation|argumentation]] is often ''winning'' at the expense of insight. Specific attitudes, beliefs, and behaviors can move us toward dialogue or away from it, toward dichotomy and fragmentation. The following table characterizes the distinctions: {| class="wikitable" |- !Toward Dialogue !! Toward Dichotomy |- | Authentic curiosity, discovery, and disclosure. Revealing information, assumptions, and doubts. Done with others. [[w:I_and_Thou|I, thou]]. || Disingenuous manipulation, secrecy, and persuasion. Disguising and defending assumptions and doubts. Maintaining distance through a polite façade or direct confrontation. Done to others. I, it. |- | Cooperation and genuine respect. Peer relationships; equality. [[Earning Trust|Trust]] and safety. [[Candor]]. Willing collaborators. || Competition, criticism, and dismissal. Displaying power; coercion. Distrust and danger. |- | Insight.|| Insult. |- | Assume Positive Intent. || Combative, competitive, malicious intent, seeking revenge, getting even, retaliation. |- | [[W:Principle of charity|Principle of charity]]. [[w:Straw_man#Steelmanning|Steelmanning]]. || Attack, [[w:Gotcha_journalism|gotcha]], [[w:Straw_man|strawman]]. |- | Listening to understand. Empathy. || Listening to respond and rebut; reloading. Apathy. |- | [[Clear_Thinking/Curriculum|Clear Thinking]]. || [[w:Rhetoric|Rhetorical Gamesmanship]]. |- | [[Intellectual Honesty|Intellectual honesty]]. || [[w:Motivated_reasoning|Motivated reasoning]]. |- | Exploring, examining, innovating, insight. Inquiry. ||Making and scoring points. I win, you lose. Advocacy. |- | Choosing to explore; inventing new ideas, creating, learning, thinking. ||Choosing to ignore; defending old postures, thoughts, and assumptions. |- | Scout mindset—Reasoning is like mapmaking. Decide what to believe by asking “Is this true?” Seek out [[Evaluating Evidence|evidence]] that will make your map more accurate.<ref>{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef|date=April 13, 2021 |title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisherPortfolio |pages=288 |isbn=978-0735217553}}</ref>||Soldier mindset—Reasoning is like defensive combat. Decide what to believe by asking either “Can I believe this?” or “Must I believe this?” depending on your motives. Seek out evidence to fortify and defend your beliefs. |- | [[Facing Facts/Reality is our common ground|Reality is our common ground]]. [[Seeking True Beliefs|Let's seek it together]].||Reality is what I say it is. Listen to me. |- | Abandoning reason is an act of violence. ||[[w:The_Art_of_Being_Right|Win at all costs]]. |- |Synthesis, combination, alternative viewpoints, integration, coherence, new possibilities. Collective intelligence. Building up, feeling constructive. [[Finding Common Ground|Finding common ground]]. ||Polarized, dichotomous thinking. Fragmentation and incoherence. Focusing on fears. Anxiety. Arrogance. Tearing down, feeling destructive. |- |[[w:Appreciative_inquiry|Appreciative inquiry]]. Shared inquiry. Seeking the strengths and possibilities in the other's ideas. Discernment. ||Criticism. Searching for flaws and weakness in the other's ideas. Judgment. |- |Deferring closure to allow complete understanding, agreement, and enduring support. ||Closing quickly to solidify your position. |- |[[Recognizing Fallacies|Identifying faulty reasoning]], information, inconsistencies, or assumptions. Willing to give up ground. ||Attacking the person. Taking ground. |- |Seeking an inclusive viewpoint; valuing and accommodating diversity. Revealing assumptions and discrepancies. || Advocating a one-sided point of view; valuing conformance. Defending a point of view and the assumptions it encompasses. |- |I can learn from you. Inclusiveness. Our doubts help to cleanse our truths. ||I am right, just listen to me. Be reasonable, do it my way. Resistance is futile. |- |[[Finding Courage|Courageous]] speech. [[Candor]]. ||Serial monologue, harangue, attacks, bloviation, obfuscation, equivocation, posturing, rehashing, gossip, small talk, party line, and idle chatter. |- |Balance of advocacy and inquiry. ||Advocacy displaces inquiry. |- |Comfortable with complexity and subtlety while seeking elegance. ||Simplistic. |- |Together we can seek the truth. Let's journey together to find it. ||I know the truth. It's my way or the highway. |- |Essence; a journey to the center of the being. Curiosity and flow. ||Image. Fear, anxiety, and [[Resolving Anger|anger]]. |- |Initial doubts leading to enduring certainty. ||Initial certainty leading to enduring doubts. |} Dialogue is more subtle and cooperative than discussion or debate. However, as a minimum, participants in dialogue must adhere to the [[w:Pragma-dialectics#Rules_for_a_critical_discussion|rules for a critical discussion]]. Another minimum standard for dialogue are [[w:Rogerian_argument#Rapoport's_rules|Rapoport's Rules]], as restated here by Daniel Dennett:<ref>{{cite book |last=Dennett |first=Daniel C. |date=May 5, 2014 |author-link=w:Daniel_Dennett |title=Intuition Pumps And Other Tools for Thinking |publisher=W. W. Norton & Company |pages=512 |isbn=978-0393348781}} Chapter 3</ref> # You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way." This is called [[w:Straw_man#Steelmanning|steelmanning]] the argument. #You should list any points of agreement (especially if they are not matters of general or widespread agreement). #You should mention anything you have learned from your target. #Only then are you permitted to say so much as a word of rebuttal or criticism. Yet another technique for discovering common ground is for each participant to answer these questions: #What about your dialogue partner's position appeals to you? #What about your own position troubles you? Please keep in mind that in dialogue the only ''target'' is insight. ===Balance Inquiry and Advocacy=== [[File:Inquiry and advocacy.jpg|350px|right|The four essential skills of dialogue.]] Dialogue requires the skillful use of four distinct practices to balance ''inquiry''—seeking to understand—and ''advocacy''—being understood. These can achieve the rhythm of respiration, first inhaling the ideas of others and later exhaling expression of your new ideas. These four skills: ''listen'', ''suspend'', ''respect'', and ''voice'' appear in the diagram on the right and are described more fully below. '''Listening to understand:''' Hear their words; [[Knowing Someone/Deep Listening|learn their meaning]]. What is the person saying? What ideas do they want to get across? What are they feeling now? What is important to them? What does this mean for them? What is not being heard? Why? What is their truth? How can I connect with them? What can I learn from them? What have I been missing? What are we all missing? How can this new information change my point of view? Who is not being heard? What are the inconsistencies, dilemmas, and paradoxes? What new frame of reference can provide coherence? Concentrate on direct observation, stick to the facts, dismiss your old thoughts and assumptions, stay in their moment, hear their story, and defer interpretation. Listen without resistance as you notice your own resistance. Notice how you are reacting. Be still; stay silent inwardly and outwardly. '''Suspending judgment:''' Defer your certainty while you explore doubt and new possibilities. Stop, step back, adopt a new point of view, and reflect from this new vantage point. Frame up—adopt a broader reference frame. Allow inquiry to displace certainty. Embrace your ignorance. Be willing to disclose your own doubts. Acknowledge what you don't know and don't understand. What am I missing? What am I protecting? Reject polarized thinking. Hold your tongue and defer forming opinions, jumping to conclusions, quick fixes, and assigning [[Attributing Blame|blame]]. Become aware of your inner reaction, but don't react outwardly. Have the discipline to hold the tension within yourself while you silently examine and reflect on it. Remain curious. Identify and examine your assumptions and theirs. Work to understand how this problem works, how has it arisen? Cope constructively with your fears and [[Resolving Anger|anger]]. Do not attribute motive or intent. Don't yet agree or disagree while you remain curious and reflect. Defer and dismiss conclusions, explore alternative meanings and motives, integrate these new ideas with the whole, and seek congruence. '''Respecting all:''' Attribute positive motives and constructive intent to each participant. Appreciate all that is good about them, all that you share in common with them, and all they can contribute. Acknowledge the dignity, legitimacy, worth, and humanity of the person speaking. Allow for differing viewpoints and learn all you can from them. Examine the origins within your self of any tendency you have to disrespect participants. Resist your temptation to [[Attributing Blame|blame]]. Remain humble and accept that they can teach us and we can learn from them. Attain and appreciate their viewpoint; do not attack, intrude, deny, dismiss, dispute, or discount their comments. Banish violence. '''Speaking your voice:''' Contribute your insight to advance the dialogue. Be patient and gather your own clear thoughts before you speak with [[candor]]; clearly, directly, and authentically. What is most important to express now? Offer your insights. Share how you feel, what you don't know, and your own doubts and concerns. Speak courageously from your own authentic voice. Avoid sarcasm, barbs, attacks, insults, reification, and condescension. Inquire and ask only genuine questions arising out of curiosity and not belligerence. Test assumptions. Speak in the first person from your actual experiences. Speak your truth. Dialogue is a dynamic process that requires a delicate balance. Inquiry—seeking new understanding—combines the skills of listening while suspending judgment to gain a deeper and newer understanding. This is balanced by advocacy—seeking to be understood. Advocacy combines respect for all participants with the courage to speak your voice, share your insights, and advance the dialogue toward a new understanding of the whole. Dialogue requires a balance between the analysis of inquiry and the action of advocacy. Inquiry and analysis alternate in balance with advocacy and action. The diagram illustrates a spiral path that encourages dialogue to emerge. Beginning with listening, we then suspend and reflect, respect others, and then speak our voice before resuming our listening. The dialogue advances the group toward the whole at the center as the participants think together. ===Dynamic Roles=== Family therapist [[w:David_Kantor|David Kantor]] describes four distinct roles that dialogue participants adopt dynamically as the dialogue proceeds: [[File:Dynamic roles during dialogue.jpg|250px|right|Dynamic roles during dialogue.]] '''Move:''' Initiate action to move the dialogue in a particular direction. Set a direction and provide clarity. '''Follow:''' Support, amplify, or derive a similar direction suggested by the preceding move. '''Oppose:''' Raise an objection to highlight possible problems or point out what may not be quite right with the current direction. '''Bystand:''' Propose a new way of thinking, a new viewpoint, a new reference frame, or a new direction that bypasses, transcends, or overcomes the temporary deadlock, expands the thinking of the group, and shows the way toward further progress. Provide perspective and encourage reflection. All four roles are required to move the dialogue along. People fill one of the roles temporarily as the conversation needs each particular type of contribution to move forward. Each role takes into account the variety of viewpoints already expressed, incorporating much of the information that has been suspended during the dialogue. The roles are dynamic, the person who ''opposed'' in one instance may ''move'' in another or ''bystand'' later on. All four roles are necessary. Without a move, there is no direction. Without the follow there is no momentum. Without the opposition, there is no critical thinking and correction, and without the bystanders, deadlocks persist and there is no breakthrough to new understanding. Conversation groups that do not achieve dialogue often get stuck in a move-oppose cycle that repeats without making progress. The maze shown at the right illustrates how the four roles work together to move the group toward the shared central understanding; the whole at the center. The ''move'' gets thing started and the ''follow'' helps keep things going. However progress seems stalled when it encounters ''opposition''. After considering all viewpoints, the ''bystander'' suggests a novel path for the group to continue along. ==Matters of Fact== [[w:Facts|Facts]] deserve a seat at the table during any dialogue. Therefore, it is important to carefully distinguish among: 1) matters of fact, 2) matters of preference, or 3) matters of controversy throughout each dialogue. Statements can be classified as one of the following three types:<ref>{{cite book |last1=Paul |first1=Richard |last2=Elder |first2=Linda |date=December 5, 2014 |title=Thinker's Guide to the Art of Socratic Questioning (Thinker's Guide Library), |publisher=Foundation for Critical Thinking |pages=134 |isbn=978-0944583319}} Three Kinds of Questions.</ref> #'''Matters of fact'''. These statements can be assessed and verified through the correct use of [[Evaluating Evidence|evidence gathering]], and reasoning. A correct statement can be made with conviction. These statements declare “what is” and careful researchers agree on the answer. Examples include: The boiling point of water is 100° Centigrade, gold is denser than lead, and the movie ''Spotlight'' won Best Picture in 2016. Notice the use of “is” to convey certainty in these statements. Do not argue matters of fact, research them instead. #'''Matters of taste, preference, or opinion'''. Any claim is acceptable here, because the statement depends only on the preferences of the person making it. Examples include: I feel that purple is the most beautiful color, I prefer chocolate ice-cream to vanilla ice-cream, and I believe that Rembrandt was a better artist than Picasso. Notice the use of “prefer”, “feel”, and “believe” to convey a personal preference. Do not argue matters of preference, enjoy them. #'''Matters of controversy'''. Although these are not opinions, sincere experts often disagree on the best answer or the best course of action. These statements propose “what ought to be” or they ask about a topic that is not yet fully and carefully explored or researched. Examples include: I believe the most pressing problem facing the world today is the lack of clean safe drinking water for all people, I think the best approach to reducing gun violence is to require comprehensive background checks for all gun purchases, and I believe incarceration rates are too high in the US. Notice the use of “believe” and “think” to convey personal positions here. Learn more about matters of controversy by exploring them with dialogue and the [[Socratic Methods|Socratic Method]]. ===Assignment=== #Read this essay on the [[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]]. #Read over this list of [[Socratic Methods/questions to classify|questions to classify]]. #Identify at least five of these questions in each of the following classifications: 1) matters of fact, 2) matters of preference, or 3) matters of controversy, #During dialogue notice the verbs used by you and your partner to convey degrees of certainly and conviction. These include: “is”, “prefer”, “feel”, “believe”, “think”, and others. Ensure the verb chosen corresponds to the degree of certainty and conviction of the statement being made. Address, explore, and correct and mismatches. #If the dialogue encounters disagreement on matters of fact, agree to research the fact and come to an agreement on the fact before continuing the dialogue. Expect you and your dialogue partner to converge on matters of fact as a result of [[w:Consilience|consilience]]. If you are unable to converge on matters of fact, begin to explore the differences in your [[Knowing_How_You_Know#What_is_a_Theory_of_Knowledge.3F|theories of knowledge]] that may be leading you toward differing conclusions. #Complete the Wikiversity course on [[Finding Common Ground]]. Find common ground. #Use agreements on matters of fact as a common basis for you and your dialogue partner to move the dialogue forward. #If the dialogue encounters disagreement on matters of taste, note the differences and continue the dialogue without requiring resolution of this disagreement. There is no correct resolution of matters of taste. #Only matters of controversy are within the useful realm of dialogue. Direct the dialogue toward these matters of controversy to move toward insights and learning together. #Complete the course on [[Facing Facts]]. ==Obstacles== Dialogue is easily spooked. There are many common obstacles that prevent dialogue from emerging. Removing sources of fear, suspending the exercise of power, eliminating external influences, removing distractions, and providing excellent communication conditions can all promote dialogue. === Fear === Fear prevents dialogue. People are often afraid to trust other participants, consider new ideas, and open up to the new possibilities that dialogue requires. People hold back and fail to participate fully and genuinely because of their fears. Suspending judgment is often an act of courage. Remaining open to new ideas; doubting, questioning, or abandoning beliefs you have held for many years, adopting a new viewpoint, releasing attachments, hearing someone for the first time, abandoning the [[w:Status quo|status quo]], thinking in a new way, allowing for change, acknowledging your old habits and beliefs, abandoning your stubbornness, admitting you don't know or don't understand, admitting you may have been wrong, exposing vulnerabilities, anticipating the ramifications and future consequences of new ideas and agreements, becoming [[w:Authenticity_(philosophy)|authentic]] rather than merely polite; and confronting assumptions, issues, and people, can all be scary. These obstacles require courage to overcome. Speaking truth to power and challenging the opinions and beliefs of others requires courage. Finding your voice requires courageous thinking. Speaking your voice requires courageous action. Have the courage to dialogue. Notice the relative salience of ''curiosity'' and ''fear''. Seek to attain and sustain curiosity. Identify and remove obstacles to curiosity. Notice when fear arises and displaces curiosity. Pause the conversation to note the emergence of fear, identify its causes, resolve the issues motivating the fear, and return to curiosity and dialogue. === External Constraints === Dialogue requires [[w:autonomy|autonomy]]. Speaking your voice requires thinking for yourself and making your own decisions. Dialogue requires adopting an internal [[w:Locus_of_control|locus of control]] and rejecting an external locus of control. Repeating the opinion of others, deferring your own judgment to someone outside the room, appealing to the views of your chosen experts or luminaries, defending a special interest, holding conflicting interests, running a secret agenda, reciting dogma, remaining star struck, going along to get along, deferring to fate or luck, or introducing external constraints such as “my boss requires . . .” or “everybody knows. . .” all prevent you from making your own decisions and speaking your own voice. Shed these external constraints so you can think for yourself, represent yourself, speak for yourself, and participate in the dialogue. Speak in the first person about your own experiences, [[w:Opinion|opinions]], and [[w:belief|beliefs]]. === Distractions === Dialogue requires focus. Multitasking seems to be emerging as the new status symbol. But dialogue is hard work that requires your full and present attention. Listening for meaning requires focus and full attention. Suspending judgment requires self discipline. Speaking your voice requires presence and thoughtfulness. Respect often requires patience and cannot be rushed. Reading mail, talking on the phone, text messaging, surfing the net, side conversations, watching the clock, preparing for the next meeting, writing notes, showboating, or wishing you were elsewhere are all distractions that will prevent you from fully participating in dialogue. Your lack of attention and concern also distracts others and may prevent them from participating in dialogue. Either focus your full and undivided attention on the conversation, or leave the room. Expect this focus of the others. === Poor Communications === Dialogue requires careful, detailed, delicate, and nuanced communications. Poor room acoustics, physical distance, language differences, accents, jargon, local vernacular, unfamiliar vocabulary, cultural differences, unshared abstractions, [[w:logical fallacies|logical fallacies]], intentional and unintentional [[w:Cognitive_distortion|distortions]], hearing difficulties, and poor sound systems can all prevent dialogue from emerging. Collocated participants in a private room free of distractions sitting comfortably in a circle where everyone can easily see and hear everyone else promotes communication that can help dialogue emerge. If language differences exist, then effective translation services, including cross-cultural translations, are required. === Bad faith actors === Practicing dialogue requires both parties to act in good faith. Work to ensure that each participant intends to do their best to remain intellectually honest and abide by the dialogue guidelines described above. However, it is difficult to predict another’s behavior until the dialogue is underway, and you may encounter tricks used in bad faith to undermine the integrity of the dialogue in an attempt to “win the argument”. One trick is called the [[w:gish gallop|Gish gallop]]. This is a rhetorical technique in which a person in a debate attempts to overwhelm their opponent by providing an excessive number of arguments with no regard for the accuracy or strength of those arguments. [[w:Mehdi_Hasan|Mehdi Hasan]] suggests using these three steps to beat the Gish gallop: <ref>Mehdi Hasan, [https://cafe.com/stay-tuned/debating-101-with-mehdi-hasan/ Stay Tuned with Preet, Debating 101], March 16, 2023. </ref> #Because there are too many falsehoods to address, it is wise to choose one as an example. Choose the one weakest, dumbest, most ludicrous argument that the opponent has presented and tear this one argument to shreds. This is called the ''weak point rebuttal''. #Don’t budge from ''this'' issue. Don’t move on from this issue until you have destroyed the nonsense and clearly and decisively made your point. #Call it out, name the strategy. “This is a strategy called the ‘Gish Gallop’, don’t be fooled by the flood of nonsense you have just heard.” Avoid entering into a debate with someone who is likely to use the Gish gallop. === Getting Unstuck === Even skillful dialogue can get stuck when It enters areas of high conflict. Journalist [[w:Amanda_Ripley|Amanda Ripley]] recommends “Complicating the Narratives”<ref> [https://thewholestory.solutionsjournalism.org/complicating-the-narratives-b91ea06ddf63 Complicating the Narratives], June 27, 2018, Solutions Journalism, Amanda Ripley.</ref> to add complexity, encourage a range of emotions to flow, allow curiosity to displace fear, invite the conversation to go deeper, and continue the dialogue. She suggests continuing the dialogue by asking one or more of these questions: *What is oversimplified about this issue? *How has this conflict affected your life? *What do you think the other side wants? *What’s the question nobody is asking? *What do you and your supporters need to learn about the other side in order to understand them better? *Tell me more. === Counterfactual Questions === Many beliefs, biases, and unkind [[w:Stereotype|stereotypes]], although strongly held, are formed arbitrarily. The arbitrary basis for many such beliefs can be exposed, and often [[w:Belief_revision|revised]] or dislodged, by posing [[w:Counterfactual_conditional|counterfactual]] questions.<ref>{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant |date= |title=Think Again: The Power of Knowing What You Don't Know|publisher=Viking|pages=320 |isbn=978-1984878106}}, Chapter 6.</ref> For example, during dialogue ask questions like these as appropriate: *If you were born in a different place, would you still hold that belief? **If you grew up in New York City instead of Boston, would you be a [[w:Yankees–Red_Sox_rivalry|Yankee’s fan rather than a Red Sox fan]]? **If you were born in a different country, would you practice a different religion? ***How would your beliefs differ? **If you were born in a different country, would your patriotism be to another country? ***Would you be fighting on the other side of this war? *Would you hold different beliefs if you: **grew up poor; **were born [[w:Disability|disabled]]; **were born another [[w:Race_(human_categorization)|race]]; **were born another sex; **were [[w:Obesity|obese]]; **were unattractive; **grew up on a farm rather than in the city; **chose different friends; **had different parents; **were an [[w:Orphan|orphan]]; **became fatally ill; **spoke a different language; **won the [[w:Lottery|lottery]]; **lived 1,000 years ago; or **worked in a different [[w:Job|job]]? How would your beliefs be different? Why would your beliefs be different? What is the basis for your current beliefs? If you can get people to pause and reflect on the origins of their beliefs, they might decide that it is irrational to apply group stereotypes to individuals, or to hold strongly to arbitrarily formed beliefs. === Recovering Dialogue === Because dialogue is fragile, conversations that begin as dialogue can shift [[Communicating Power|their tone]] and become argumentative. Take care to notice when this happens, pause the conversation to announce this shift and restore the dialogue. The shift often occurs when curiosity yields to fear. Throughout each dialogue session: #Periodically notice the [[Communicating Power|tone of the conversation]] to determine if the rules of dialogue are being followed. #If you notice a slip away from dialogue toward debate or other argumentative forms of communication, interrupt and pause the conversation. A request such as “Let’s take a break here to examine our dialogue skills in action” may provide an effective transition. #Describe the shift in conversational tone that you have noticed. It may be helpful to identify [[Practicing_Dialogue#Toward_Dialogue|specific dialogue characteristics]] that are missing, or the argumentative characteristics that have appeared. This conversation may be contentious. Cite specific examples such as “When you said X, were you speaking from curiosity or from power? (pause here to allow a thoughtful answer) Were you seeking insight or trying to win an argument? (pause) Were you assuming my positive intent, or becoming combative?” #Ask that these argumentative behaviors be abandoned. Resolve to return to the goal of ''insight'' rather than ''winning''. Allow this topic to be discussed if this helps identify the characteristics that distinguish dialogue from argumentation. It may be helpful to explore the prevailing [[Socratic_Methods#Essential_Socratic_Temperament|temperaments of the participants]] to determine if conditions are suitable for continuing dialogue. #It may be helpful to take a short break to reflect on what happened, improve your dialogue skills, strengthen your relationship, reaffirm your intent to continue dialogue, and sharpen your ability to notice when and how the dialogue transformed. #Resume the dialogue from some place before it became argumentative. #Continue to monitor the tone of communication and pause if argumentation arises again. #Use these [[/Phrases for managing conversations/]] throughout the process. == Success Stories == The power of dialogue has achieved some successful solutions to very difficult problems. Here are some examples: * The [http://compact.org/resource-posts/the-san-diego-dialogue/ San Diego Dialogue project] is contributing to the advancement of research, relationships and solutions to the San Diego-Baja California crossborder region's long-term challenges in innovation, economy, health and education. * The [http://ncdd.org/ National Coalition for Dialogue and Deliberation] works to give people a voice in important issues. Their website documents many successful dialogue projects. * [[w:Vicki_Robin#Conversation_Caf.C3.A9|Conversation Café]] groups are improving conversations and strengthening the interconnections among people across America. *The [http://www.civilconversationsproject.org/ Civil Conversations Project] seeks to renew common life in a fractured and tender world. * [https://livingroomconversations.org Living Room Conversations] are a conversational bridge across issues that divide and separate us. ==Assignment== # Learn the [[Practicing_Dialogue#Toward_Dialogue|distinctive skills of dialogue]]. # Remove the [[Practicing_Dialogue#Obstacles|obstacles described above]]. # Assemble and engage the stakeholders. # Create the space, increase safety, [[Earning Trust|build trust]], level power, defer decision making, demonstrate empathy. # Invite the group to do something truly important, and then # stand back. Allow an important dialogue to emerge as meaning begins to flow. ==Optional Assignment== # Read the essay [[/From Demagoguery to Dialogue/]] # Stay alert for opportunities to intervene in a discussion gone bad, stop the action, ask for fact checking, remind the discussants of the rules for dialogue, and encourage the discussants to practice dialogue. == Further Reading == Students interested in learning more about dialogue may be interested in the following materials: *{{cite book |last=Yankelovich |first=Daniel |date=September 5, 2001 |title=The Magic of Dialogue: Transforming Conflict into Cooperation |publisher= Touchstone |pages=240 |isbn=978-0684865669}} *{{cite book |last=Isaacs |first=William |date=September 14, 1999 |title=Dialogue: The Art Of Thinking Together |publisher=Crown Business |pages=448 |isbn=978-0385479998}} *{{cite book |last=Bohm |first=David |date= |title=On Dialogue |publisher=Routledge |pages=144 |isbn=978-0415336413}} *{{cite book |last=de Bono |first=Edward |date=August 18, 1999 |title=[[w:Six_Thinking_Hats|Six Thinking Hats]] |publisher=Back Bay Books |pages=192 |isbn=978-0316178310}} *{{cite book |last=Runion |first=Meryl |date=December 31, 2003 |title=How to Use Power Phrases to Say What You Mean, Mean What You Say, & Get What You Want |publisher=McGraw-Hill Education |pages=224 |isbn=}} *{{cite book |last1=Fisher |first1=Roger |last2=Ury |first2=William L. |date=December 1, 1991 |title=[[w:Getting_to_Yes|Getting to Yes]]: Negotiating Agreement Without Giving In |publisher=Penguin Books |pages=200 |isbn=978-0140157352}} *{{cite book |last=Ury |first=William |date=February 27, 2007 |title=The Power of a Positive No: How to Say No and Still Get to Yes |publisher=Bantam |pages=272 |isbn=978-0553804980}} *{{cite book |last1=Fisher |first1=Roger |last2=Shapiro |first2=Daniel |date=October 6, 2005 |title=Beyond Reason: Using Emotions as You Negotiate |publisher=Viking Adult |pages=256 |isbn=978-0670034505}} *{{cite book |last1=Miller |first1=William R. |last2=Rollnick |first2=Stephen |date=April 12, 2002 |title=[[w:Motivational_interviewing|Motivational Interviewing]]: Preparing People for Change, 2nd Edition |publisher=The Guilford Press |pages=428 |isbn=978-1572305632}} *{{cite book |last=Frankfurt |first=Harry G. |date=January 30, 2005 |title=[[w:On_Bullshit|On Bullshit]] |publisher=Princeton University Press |pages=67 |isbn= 978-0691122946}} *{{cite book |last=Galtung |first=Johan |date=July 1, 2004 |title=Transcend and Transform: An Introduction to Conflict Work |publisher=Routledge |pages=200 |isbn=978-1594510632}} *{{cite book |last=Friedman |first=Maurice S. |date=November 10, 2002 |title=Martin Buber: The Life of Dialogue |publisher=Routledge |pages=432 |isbn=978-0415284752}} *{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef|date=April 13, 2021 |title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisherPortfolio |pages=288 |isbn=978-0735217553}} *{{cite book |last=Briskin |first=Alan |date=October 1, 2009 |title=The Power of Collective Wisdom: And the Trap of Collective Folly |publisher=Berrett-Koehler Publishers |pages=220 |isbn=978-1576754450}} *{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant |date= |title=Think Again: The Power of Knowing What You Don't Know|publisher=Viking|pages=320 |isbn=978-1984878106}} * [https://www.ted.com/talks/robb_willer_how_to_have_better_political_conversations How to have better political conversations], TED Talk, September 2016, Robb Willer * [http://www.npr.org/programs/ted-radio-hour/558307433 Dialogue And Exchange], TED Radio Hour program, Friday October 27, 2017 * [https://www.ted.com/talks/joan_blades_and_john_gable_free_yourself_from_your_filter_bubbles?language=en Free yourself from your filter bubbles], TED Talk, November 2017, Joan Blades and John Gable I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. *[[w:The_Art_of_Being_Right|''The Art of Always Being Right'']]: ''Thirty Eight Ways to Win When You Are Defeated'', by Grayling, A. C. ==References== <references/> {{Emotional Competency}} {{CourseCat}} [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Peace studies]] [[Category:Humanities courses]] [[Category:Community]] [[Category:Social Skills]] 7datoxh2uky0dka7oq5ofv31ms4rp3z 0 212733 2691771 2691542 2024-12-13T09:24:15Z Young1lim 21186 /* Using Libraries */ 2691771 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20241213.pdf |pdf]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] 1s8fuqrwp5ig13ovpawkuw6qnijatfe 10 213215 2691634 2691605 2024-12-12T14:42:46Z Atcovi 276019 Reverted edits by [[Special:Contributions/38.51.153.20|38.51.153.20]] ([[User_talk:38.51.153.20|talk]]) to last version by [[User:Ear-phone|Ear-phone]] using [[Wikiversity:Rollback|rollback]] 2175055 wikitext text/x-wiki <noinclude>{{WikiJMed top menu}}{{WikiJMed right menu}} '''In the news''' </noinclude><includeonly><div style="float:right"><small>[{{edit|Template:WikiJMed right menu/In the news}}]</small></div></includeonly> '''2020:''' <br> [[:en:w:Wikipedia:Wikipedia_Signpost/2020-06-28/News_and_notes|Article in ''The Signpost'']] '''2019:''' <br> [https://drive.google.com/file/d/1ZQ3AsK1H_kAYuJQ8082xCuPgQyYmJDje/view?usp=sharing Presentation at Wikimania 2019] <br> [[:en:File:WikiJournals_for_psychologists_(APA2019).pdf|Presentation at American Psychological Association]] <br> [[:en:w:Wikipedia:Wikipedia Signpost/2019-06-30/In focus|Article in ''The Signpost'']] <br> [https://blog.wikimedia.org.uk/2019/03/what-are-wikijournals/ Article in the Wikimedia UK blog] '''2018:''' <br>[https://youtu.be/PB0wl8Mi4Bw?t=356 Presentation] at [https://blog.apo.org.au/2018/12/14/apo-forum-2018-videos-and-slides/ APO forum] <br>[https://wikimediafoundation.org/2018/09/24/wikijournal-interview/ Interview with Wikimedia Foundation] <br>[https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Best_articles_of_2017 Winners of the best articles submitted in 2017] <br>[[Wikipedia:Committee on Publication Ethics|Gained COPE membership]] <br>[https://www.youtube.com/watch?v=WiXkLN3phYc Presentation] at [https://wikimania2018.wikimedia.org/wiki/Programme/Wiki_Project_Med Wikimania 2018] <includeonly>''[[WikiJournal of Medicine/In news and media|More mentions]]''</includeonly><noinclude><!-- -->'''2017:''' <br>[https://www.statnews.com/2017/10/10/updating-wikipedia-part-doctors-jobs/ Article in ''Stat news''] <br>[http://jech.bmj.com/content/early/2017/08/24/jech-2016-208601 Article in ''Journal of Epidemiology and Public Health''] <br>[http://science.sciencemag.org/content/357/6351/557.2.long Article in ''Science''] <br>[https://www.youtube.com/watch?v=WKIp-cFSKvE Interview with board member] '''2016:''' <br>[https://theconversation.com/wikipedia-is-already-the-worlds-dr-google-its-time-for-doctors-and-researchers-to-make-it-better-66769 Article 1 in ''The Conversation''] <br>[https://theconversation.com/why-getting-medical-information-from-wikipedia-isnt-always-a-bad-idea-59708 Article 2 in ''The Conversation''] <br>[https://sourcecode.berlin/2016/06/22/wikipedia-doctors-and-the-future-of-medicine/#t=25:39.258 Interview featuring Editor-in-chief] <br>[https://soundcloud.com/primediabroadcasting/why-getting-medical-information-from-wikipedia-isnt-always-a-bad-idea Interview featuring Assistant to the editor-in-chief] <br>[[Wikipedia:Wikipedia:Wikipedia_Signpost/2016-06-15/Special_report|Article in ''The Signpost'']] disqch4b3uju0n1kxu3b88xc2viid2l 0 216855 2691708 2636911 2024-12-12T23:00:34Z Yinuo Liu 2935880 delete extra spaces 2691708 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{Wikipedia2|General Behavior Inventory}} {{contrib-major|Eyoungstrom}} The '''[[w:General Behavior Inventory (GBI)|General Behavior Inventory (GBI)]]''' is a questionnaire designed by Richard Depue and colleagues to measure [[w:Mania|manic]] and [[w:depression_(mood)|depressive]] symptoms in adults, as well as to assess for [[w:Cyclothymia|cyclothymic disorder]].<ref name=":1" /> It is one of the most widely used psychometric tests for measuring the severity of bipolar disorder in research, and it also can track the fluctuation of symptoms over time. The GBI was first designed for use with adults; however, it has been adapted into versions that allow parents to rate their children. Multiple short versions are available that are often more convenient to use. Current versions include the original full length GBI (self-report), the '''Parent GBI (P-GBI)''', the '''Parent GBI-10-item Mania Scale (P-GBI-10M)''' and two '''Parent GBI-10-item Depression Scales (Form A & B)''', and the '''7 Up-7 Down Inventory''', as well as a '''Sleep''' scale. A version was tested as a teacher-report version, but it proved not to be practical for general use.<ref>{{Cite journal|last1=Youngstrom|first1=Eric A.|last2=Joseph|first2=Megan F.|last3=Greene|first3=Jamelle|date=2008|title=Comparing the psychometric properties of multiple teacher report instruments as predictors of bipolar disorder in children and adolescents|journal=Journal of Clinical Psychology|volume=64|issue=4|pages=382–401|doi=10/cjbtds|issn=1097-4679}}</ref>[[File:C. Graham Ford coding General Behavior Inventory data.jpg|thumb|A research volunteer rebuilding a data table from a scatterplot in one of the first publications using the GBI]] == Access and Use == The General Behavior Inventory is free to use in both research and clinical work. Some of the short forms have been formally CC-BY-SA licensed. The original author, Richard Depue, asks that you contact him to let him know about the project (rad5@cornell.edu). The GBI has been translated into several languages (in short forms as well as full length, and parent as well as self-report); see the '''External Links''' section for a link to an extensive repository. Suggested citations are: ===== Self report, full length version ===== Depue, R. A., Slater, J. F., Wolfstetter-Kausch, H., Klein, D. N., Goplerud, E., & Farr, D. A. (1981). A behavioral paradigm for identifying persons at risk for bipolar depressive disorder: A conceptual framework and five validation studies. ''Journal of Abnormal Psychology, 90,'' 381-437. <nowiki>https://doi.org/10.1037/0021-843X.90.5.381</nowiki><ref>{{Cite journal|last=Depue|first=Richard A.|last2=Slater|first2=Judith F.|last3=Wolfstetter-Kausch|first3=Heidi|last4=Klein|first4=Daniel|last5=Goplerud|first5=Eric|last6=Farr|first6=David|date=1981|title=A behavioral paradigm for identifying persons at risk for bipolar depressive disorder: A conceptual framework and five validation studies.|url=http://doi.apa.org/getdoi.cfm?doi=10.1037/0021-843X.90.5.381|journal=Journal of Abnormal Psychology|language=en|volume=90|issue=5|pages=381–437|doi=10.1037/0021-843X.90.5.381|issn=1939-1846}}</ref> and if using specifically in teens: Danielson, C. K., Youngstrom, E. A., Findling, R. L., & Calabrese, J. R. (2003). Discriminative validity of the General Behavior Inventory using youth report. ''Journal of Abnormal Child Psychology, 31,'' 29-39. ===== Parent report about youth symptoms (full length version): ===== Youngstrom, E. A., Findling, R. L., Danielson, C. K., & Calabrese, J. R. (2001). Discriminative validity of parent report of hypomanic and depressive symptoms on the General Behavior Inventory. ''Psychological Assessment, 13,'' 267-276. ===== Teacher report (not recommended for clinical use): ===== Youngstrom, E. A., Joseph, M. F., & Greene, J. (2008). Comparing the psychometric properties of multiple teacher report instruments as predictors of bipolar disorder in children and adolescents. ''Journal of Clinical Psychology, 64,'' 382-401. <nowiki>http://dx.doi.org/10.1002/jclp.20462</nowiki><ref>{{Cite journal|last=Youngstrom|first=Eric A.|last2=Joseph|first2=Megan F.|last3=Greene|first3=Jamelle|date=2008-04|title=Comparing the psychometric properties of multiple teacher report instruments as predictors of bipolar disorder in children and adolescents|url=http://doi.wiley.com/10.1002/jclp.20462|journal=Journal of Clinical Psychology|language=en|volume=64|issue=4|pages=382–401|doi=10.1002/jclp.20462}}</ref> ===== Sleep scale ===== Meyers, O. I., & Youngstrom, E. A. (2008). A Parent General Behavior Inventory subscale to measure sleep disturbance in pediatric bipolar disorder. ''Journal of Clinical Psychiatry, 69,'' 840-843. <nowiki>https://doi.org/ej07m03594</nowiki> ===== 10 Item Mania and Depression forms ===== ====== Parent report ====== Youngstrom, E. A., Van Meter, A. R., Frazier, T. W., Youngstrom, J. K., & Findling, R. L. (2018). Developing and validating short forms of the Parent General Behavior Inventory Mania and Depression Scales for rating youth mood symptoms. ''Journal of Clinical Child & Adolescent Psychology.'' <nowiki>https://doi.org/https://doi.org/10.1080/15374416.2018.1491006</nowiki><ref>{{Cite journal|last=Youngstrom|first=Eric A.|last2=Van Meter|first2=Anna|last3=Frazier|first3=Thomas W.|last4=Youngstrom|first4=Jennifer Kogos|last5=Findling|first5=Robert L.|date=2020-03-03|title=Developing and Validating Short Forms of the Parent General Behavior Inventory Mania and Depression Scales for Rating Youth Mood Symptoms|url=https://www.tandfonline.com/doi/full/10.1080/15374416.2018.1491006|journal=Journal of Clinical Child & Adolescent Psychology|language=en|volume=49|issue=2|pages=162–177|doi=10.1080/15374416.2018.1491006|issn=1537-4416}}</ref> ====== Self report ====== Youngstrom, E. A., Perez Algorta, G., Youngstrom, J. K., Frazier, T. W., & Findling, R. L. (2020). Evaluating and Validating GBI Mania and Depression Short Forms for Self-Report of Mood Symptoms. Journal of Clinical Child & Adolescent Psychology, 1-17. ===== 7 Up-7 Down ===== Youngstrom, E. A., Murray, G., Johnson, S. L., & Findling, R. L. (2013). The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory. ''Psychological Assessment, 25,'' 1377-1383. <nowiki>https://doi.org/10.1037/a0033975</nowiki><ref>{{Cite journal|last=Youngstrom|first=Eric A.|last2=Murray|first2=Greg|last3=Johnson|first3=Sheri L.|last4=Findling|first4=Robert L.|date=2013|title=The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory.|url=http://doi.apa.org/getdoi.cfm?doi=10.1037/a0033975|journal=Psychological Assessment|language=en|volume=25|issue=4|pages=1377–1383|doi=10.1037/a0033975|issn=1939-134X|pmc=PMC3970320|pmid=23914960}}</ref> Note that the 7 Up has less content coverage, and small but significant differences in reliability and validity compared to the 10 item mania scale. Another practical (and sometimes ethical) consideration is that the 7 Down includes the suicidal ideation item, whereas the 10 item depression short forms do not ask about suicidal ideation. Other short forms have not been extensively published or replicated. == Scoring and interpretation == === GBI scoring === The current GBI questionnaire includes 73 Likert-type items which reflect symptoms of different moods, and six additional validity items at the end. The original version of the GBI used "case scoring" where items were coded as "threshold" or "not at threshold." Symptoms that were rated as 1 or 2 were considered to be absent and symptoms rated as 3 or 4 were considered to be present. However, Likert scaling would be a better scoring option. The items on the GBI are now scaled from 0-3 rated as 0 (never or hardly ever present), 1 (sometimes present), 2 (often present), and 3 (very often or almost constantly present).<ref name="ReferenceA">{{cite journal|last1=Findling|first1=RL|last2=Youngstrom|first2=EA|last3=Danielson|first3=CK|last4=DelPorto-Bedoya|first4=D|last5=Papish-David|first5=R|last6=Townsend|first6=L|last7=Calabrese|first7=JR|title=Clinical decision-making using the General Behavior Inventory in juvenile bipolarity.|journal=Bipolar disorders|date=February 2002|volume=4|issue=1|pages=34–42|pmid=12047493}}</ref> All versions that we are circulating now use the 0 to 3 anchors. The scoring for the self-report and parent report versions are the same for the full length, sleep, and 10 and 7 item short forms. The ''Depression Scale'' consists of the sum of items: 01, 03, 05, 06, 09, 10, 12, 13, 14, 16, 18, 20, 21, 23, 25, 26, 28, 29, 32, 33, 34, 36, 37, 39, 41, 44, 45, 47, 49, 50, 52, 55, 56, 58, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73. The ''Hypomanic/Biphasic'' scale sums these items: 02, 04, 07, 08, 11, 15, 17, 19, 22, 24, 27, 30, 31, 35, 38, 40, 42, 43, 44, 46, 48, 51, 53, 54, 57, 61, 64, 66. (Note that Depue's scoring includes item 44 on both the depression and the hypomanic/biphasic scale). To compare scores to data offered by Youngstrom et al. (any publication), just add the items (if scored 0 to 3), or add the items and subtract the number of items on the scale (if scored 1 to 4). To compare scores to college student and adult data (published by Depue, Klein, and others), check carefully whether to use the 0/1 case scoring method versus a form of Likert scoring, as appropriate. === GBI Short Forms === [[File:Content mapping for GBI parcels onto the 7 Up-7 Down scales.png|thumb|The 7 Up-7 Down has relatively narrow content coverage]] There are several short forms that have been carved from the GBI. These include the 10-item mania and depression scales<ref>{{Cite journal|last1=Youngstrom|first1=E. A.|last2=Van Meter|first2=A.|last3=Frazier|first3=T. W.|last4=Youngstrom|first4=J. K.|last5=Findling|first5=R. L.|date=2018-07-24|title=Developing and Validating Short Forms of the Parent General Behavior Inventory Mania and Depression Scales for Rating Youth Mood Symptoms|journal=Journal of Clinical Child & Adolescent Psychology|pages=1–16|doi=10/gdvntr|issn=1537-4424 (Electronic) 1537-4416 (Linking)}}</ref>, as well as the 7-item versions (the 7 Up-7 Down scale)<ref>{{Cite journal|last1=Youngstrom|first1=E. A.|last2=Murray|first2=G.|last3=Johnson|first3=S. L.|last4=Findling|first4=R. L.|date=2013-12|title=The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory|journal=Psychological Assessment|volume=25|issue=4|pages=1377–83|doi=10/f5kp9c|issn=1939-134X (Electronic) 1040-3590 (Linking)}}</ref>, and a sleep scale<ref>{{Cite journal|last1=Meyers|first1=O. I.|last2=Youngstrom|first2=E. A.|date=2008-05|title=A Parent General Behavior Inventory subscale to measure sleep disturbance in pediatric bipolar disorder|journal=Journal of Clinical Psychiatry|volume=69|issue=5|pages=840–3|doi=ej07m03594 [pii]|issn=1555-2101 (Electronic)}}</ref>. For all of these, the score is simply the sum of the items. If these are given as standalone scales, the the 10 item scales are the sum of the ten items. If the 73-item version is given, then these are the items to extract for each short form: * ''Sleep'' (7 items): 5, 15, 18, 25, 31, 37, 52. ** 5 and 18 load on a lassitude factor, and 52 crossloads ** 15, 25, 31, 37 and 52's main loading are on the larger insomnia factor * ''10 item Mania'': 53, 54, 4, 11, 22, 40, 27, 19, 64, 31. * ''10 item Depression Form A'': 3, 45, 68, 16, 56, 13, 5, 20, 50, 59. * ''10 item Depression Form B'': 34, 14, 63, 72, 62, 9, 23, 6, 32, 18). * ''7 Down'': 23, 34, 63, 47, 56, 62, 73. * ''7 Up'': 22, 31, 30, 64, 43, 46, 38. [[File:Content mapping for GBI parcels--10 item Mania and Depression forms.png|thumb|The 10 item short forms cover more facets of mood symptoms]] * ''Hypomania'' (Depue's scoring): 4, 7, 8, 11, 15, 17, 22, 27, 30, 31, 38, 42, 43, 44, 46, 51, 54, 57, 61, 64, 66. * ''Mixed'' (Depue's scoring): 2, 19, 24, 35, 40, 48, 53). Note that researchers have used other sets of items as short forms. A sortable table below shows the overlap between items across the different forms, along with the item content: Additional research versions: * ''LAMS 12 Item'' self report: 52, 40, 44, 59, 19, 29 (factor 1); 11, 7, 31, 38, 22, 4 (factor 2). * Lewinsohn used a 12 item version in the Oregon epidemiological study: '''4''', 8, '''11''', 15, 30, 44, 51, '''64''' (all from Depue's hypomanic set); 2, '''19''', 24, 48 (from Depue's mixed set). Boldfaced items overlap with the 10M scale and could be used in a calibration study. * Jensen et al. rationally derived a 7 item impulsive aggression scale: 27, 42, 44, 51, 14, 39, 53, 54. {| class="wikitable sortable mw-collapsible mw-collapsed" |- !'''Item #''' !'''Variable''' !'''Depue 2 scale''' !'''Depue 3 scale''' !'''7Up-7D''' !'''Sleep''' !'''10 item Forms''' !LAMS 12 !Impulsive Aggression !'''Parcel'''<ref>{{Cite journal|last=Youngstrom|first=Eric A.|last2=Findling|first2=Robert L.|last3=Danielson|first3=Carla Kmett|last4=Calabrese|first4=Joseph R.|date=2001-06|title=Discriminative validity of parent report of hypomanic and depressive symptoms on the General Behavior Inventory.|url=https://doi.apa.org/doi/10.1037/1040-3590.13.2.267|journal=Psychological Assessment|language=en|volume=13|issue=2|pages=267–276|doi=10.1037/1040-3590.13.2.267|issn=1939-134X}}</ref> !'''Parcel Label''' !'''Content''' |- |22 |agbi22 |HB |hyp |u | |M |F2 | |4 |elated mood |Has your child had periods of extreme happiness and intense energy lasting several days or more when he/she also felt much more anxious or tense (jittery, nervous, uptight) than usual (other than related to the menstrual cycle)? |- |31 |agbi31 |HB |hyp |u |sleep |M |F2 | |4 |elated mood |Has your child had periods of extreme happiness and intense energy (clearly more than his/her usual self) when, for several days or more, it took him/her over an hour to get to sleep at night? |- |64 |agbi64 |HB |hyp |u | |M | | |7 |racing thoughts, cog disturb (up) |Has your child had times when his/her thoughts and ideas came so fast that he/she couldn’t get them all out, or they came so quickly others complained that they couldn’t keep up with your child ideas? |- |4 |agbi04 |HB |hyp | | |M |F2 | |3 |Increased energy |Has your child experienced periods of several days or more when, although he/she was feeling unusually happy and intensely energetic (clearly more than your child’s usual self), he/she was also physically restless, unable to set still, and had to keep moving or jumping from one activity to another? |- |11 |agbi11 |HB |hyp | | |M |F2 | |5 |high drive |Have there been periods of several days or more when your child’s friends or other family members told you that your child seemed unusually happy or high – clearly different from his/her usual self or from a typical good mood? |- |19 |agbi19 |HB |mix | | |M |F1 | |2 |Mood never in middle |Has your child’s mood or energy shifted rapidly back and forth from happy to sad or high to low? |- |27 |agbi27 |HB |hyp | | |M | |ImpAgg |6 |rage, manic irritability |Have there been times of several days or more when, although your child was feeling unusually happy and intensely energetic (clearly more than his/her usual self), he/she also had to struggle very hard to control inner feelings of rage or an urge to smash or destroy things? |- |40 |agbi40 |HB |mix | | |M |F1 | |2 |Mood never in middle |Have you found that your child’s feelings or energy are generally up or down, but rarely in the middle? |- |53 |agbi53 |HB |mix | | |M | |ImpAgg |2 |Mood never in middle |Has your child had periods lasting several days or more when he/she felt depressed or irritable, and then other periods of several days or more when he/she felt extremely high, elated, and overflowing with energy? |- |54 |agbi54 |HB |hyp | | |M | |ImpAgg |6 |rage, manic irritability |Have there been periods when, although your child was feeling unusually happy and intensely energetic, almost everything got on his/her nerves and make him/her irritable or angry (other than related to the menstrual cycle? |- |23 |agbi23 |dep |dep |d | |Db | | |9 |feels sad |Have there been times of several days or more when your child was so sad that it was quite painful for him/her, or he/she felt that he/she couldn’t stand it? |- |34 |agbi34 |dep |dep |d | |Db | | |18 |rumination(?) |Over the past year, have there been long periods in your child’s life when he/she felt sad, depressed, or irritable most of the time? |- |62 |agbi62 |dep |dep |d | |Db | | |10 |hopeless, low self-esteem |Has your child had periods when it seemed that the future was hopeless and things could not improve? |- |63 |agbi63 |dep |dep |d | |Db | | |9 |feels sad |Have there been periods lasting several days or more when your child was so down in the dumps that he/she thought he/she might never snap out of it? |- |6 |agbi06 |dep |dep | | |Db | | |20 |tearful, sad appearance |Have people said that your child looked sad or lonely? |- |9 |agbi09 |dep |dep | | |Db | | |11 |loss of interest |Have there been periods lasting several days or more when your child lost almost all interest in people close to him/her and spent long times by himself/herself? |- |14 |agbi14 |dep |dep | | |Db | |ImpAgg |15 |dep irritable mood |Has your child had periods of sadness and depression when almost everything gets on his/her nerves and makes him/her irritable or angry (other than related to the menstrual cycle)? |- |18 |agbi18 |dep |dep | |sleep |Db | | |19 |atypical dep features |Have there been times of several days or more when your child was so tired and worn out that it was very difficult or even impossible to do his/her normal everyday activities (not including times of intense exercise, physical illness, or heavy work schedules)? |- |32 |agbi32 |dep |dep | | |Db | | |18 |rumination |Over the past year, have there been times when your child looked back over his/her life and could see only failures or hardships? |- |72 |agbi72 |dep |dep | | |Db | | |18 |rumination |Have there been periods of time when your child felt a persistent sense of gloom? |- |56 |agbi56 |dep |dep |d | |Da | | |10 |hopeless, low self-esteem |Have there been times of several days or more when your child really got down on himself/herself and felt worthless? |- |3 |agbi03 |dep |dep | | |Da | | |9 |feels sad |Has your child become sad, depressed, or irritable for several days or more without really understanding why? |- |5 |agbi05 |dep |dep | |sleep |Da | | |14 |sleep disturb (dep) |Have there been periods of several days or more when your child felt he/she needed more sleep, even though he/she slept longer at night or napped more during the day (not including times of exercise, physical illness, or heavy work schedules)? |- |13 |agbi13 |dep |dep | | |Da | | |11 |loss of interest |Have there been times when your child lost almost all interest in the things that he/she usually likes to do (such as hobbies, school, work, entertainment)? |- |16 |agbi16 |dep |dep | | |Da | | |17 |somatic sx (appetite & sleep) |Has your child had long periods in which he/she felt that he/she couldn’t enjoy life as easily as other people? |- |20 |agbi20 |dep |dep | | |Da | | |18 |rumination |Have there been periods lasting several days or more when your child spent much of his/her time brooding about unpleasant things that have happened? |- |45 |agbi45 |dep |dep | | |Da | | |9 |feels sad |Over the past year, have there been times of several days or more when your child was so down that nothing (not even friends or good news) could cheer him/her up? |- |50 |agbi50 |dep |dep | | |Da | | |16 |excess guilt & paranoia |Has your child had sad and depressed periods lasting several days or more when he/she also felt much more anxious or tense (jittery, nervous, uptight) than usual (other than related to the menstrual cycle)? |- |59 |agbi59 |dep |dep | | |Da |F1 | |12 |low energy/anhedonia |Have there been periods of several days or more when your child was slowed down and couldn’t move as quickly as usual? |- |68 |agbi68 |dep |dep | | |Da | | |19 |atypical dep features |Has your child had long periods when he/she was down and depressed, interrupted by brief periods when his/her mood was normal or slightly happy? |- |30 |agbi30 |HB |hyp |u | | | | |4 |elated mood |Have there been times lasting several days or more when your child felt he/she must have lots of excitement, and he/she actually did a lot of new or different things? |- |38 |agbi38 |HB |hyp |u | | |F2 | |8 |grandiosity |Has your child had periods of extreme happiness and high energy lasting several days or more when what your child saw, heard, smelled, tasted, or touched seemed vivid or intense? |- |43 |agbi43 |HB |hyp |u | | | | |8 |grandiosity |Have there been periods of several days or more when your child’s thinking was so clear and quick that it was much better than most other people’s? |- |46 |agbi46 |HB |hyp |u | | | | |8 |grandiosity |Have there been times of several days or more when your child felt that he/she was a very important person or that his/her abilities or talents were better than most other people’s? |- |47 |agbi47 |dep |dep |d | | | | |10 |hopeless, low self-esteem |Have there been times when your child hated himself/herself or felt that he/she was stupid, ugly, unlovable, or useless? |- |73 |agbi73 |dep |dep |d | | | | |10 |hopeless, low self-esteem |Have there been times when your child felt that he/she would be better off dead? |- |1 |agbi01 |dep |dep | | | | | |13 |cog disturb (dep) |Have there been periods in your child’s life over the past year when it was almost impossible to make small decisions even though this may not be generally true of him/her? |- |2 |agbi02 |HB |mix | | | | | |1 |Extremes  of Mood & Energy |Have you found your child’s enjoyment in being with people changes -- from times when he/she enjoys them immensely and wants to be with them all the time, to times when he/she does not want to see them at all? |- |7 |agbi07 |HB |hyp | | | |F2 | |3 |Increased energy |Have there been periods of several days or more when your child was almost constantly active such that others told you they couldn’t keep up with him/her or that he/she wore them out? |- |8 |agbi08 |HB |hyp | | | | | |7 |racing thoughts, cog disturb (up) |Have there been periods of several days or more when your child could not keep his/her attention on any one thing for more than a few seconds, and his/her mind jumped rapidly from one thought to another or to things around him/her? |- |10 |agbi10 |dep |dep | | | | | |11 |loss of interest |Has your child had periods of several days or more when food seemed rather flavorless and he/she didn’t enjoy eating at all? |- |12 |agbi12 |dep |dep | | | | | |13 |cog disturb (dep) |Have there been times when your child’s memory or concentration seemed especially poor and he/she found it difficult, for example, to read or follow a TV program, even though he/she tried? |- |15 |agbi15 |HB |hyp | |sleep | | | |3 |Increased energy |Have there been times of several days or more when your child did not feel the need for sleep and was able to stay awake and alert for much longer than usual because he/she was full of energy? |- |17 |agbi17 |HB |hyp | | | | | |5 |high drive |Has your child had periods of several days or more when he/she wanted to be with people so much of the time that they asked your child to leave them alone for awhile? |- |21 |agbi21 |dep |dep | | | | | |12 |low energy/anhedonia |Have there been times when your child felt that he/she was physically cut off from other people or from himself/herself, or felt as if he/she was in a dream, or felt that the world looked different or had changed in some way? |- |24 |agbi24 |HB |mix | | | | | |1 |Extremes  of Mood & Energy |Have you found that your child’s enjoyment in eating changes – from periods of two or more days when food tastes exceptionally good, clearly better than usual, to other periods of several days or more when food seems rather flavorless and perhaps your child doesn’t enjoy eating at all? |- |25 |agbi25 |dep |dep | |sleep | | | |14 |sleep disturb (dep) |Have there been times of several days or more when your child wakes up much too early in the morning and has problems getting back to sleep? |- |26 |agbi26 |dep |dep | | | | | |19 |atypical dep features |Has your child had periods when he/she was so down that he/she found it hard to start talking or that talking took too much energy? |- |28 |agbi28 |dep |dep | | | | | |20 |tearful, sad appearance |Have there been periods other than when your child was physically ill that he/she had more than one of the following: |- |29 |agbi29 |dep |dep | | | |F1* | |16 |excess guilt & paranoia |Has your child experienced periods of several days or more when he/she was feeling down and depressed, and he/she also was physically restless, unable to sit still, and had to keep moving or jumping from one activity to another? |- |33 |agbi33 |dep |dep | | | | | |12 |low energy/anhedonia |Has your child experienced times of several days or more when he/she felt as if he/she was moving in slow motion? |- |35 |agbi35 |HB |mix | | | | | |1 |Extremes  of Mood & Energy |Has it seemed that your child experiences both pleasurable and painful emotions more intensely than other people? |- |36 |agbi36 |dep |dep | | | | | |16 |excess guilt & paranoia |Have there been periods of several days or more when your child felt guilty and thought he/she deserved to be punished for something he/she had or had not done? |- |37 |agbi37 |dep |dep | |sleep | | | |14 |sleep disturb (dep) |Has your child had times of several days or more when he/she woke up frequently or had trouble staying asleep during the middle of the night? |- |39 |agbi39 |dep |dep | | | | |ImpAgg |15 |dep irritable mood |Have there been times when your child was feeling low and depressed, and he/she also had to struggle very hard to control inner feelings of rage or an urge to smash or destroy things? |- |41 |agbi41 |dep |dep | | | | | |13 |cog disturb (dep) |Has your child had periods of several days or more when it was difficult or almost impossible to think and his/her mind felt sluggish, stagnant, or “dead”? |- |42 |agbi42 |HB |hyp | | | | |ImpAgg |5 |high drive |Have there been times when your child had a strong urge to do something mischievous, destructive, risky, or shocking? |- |44 |agbi44 |dep |dep | | | |F1 |ImpAgg |6 |rage, manic irritability |Have there been times when your child exploded at others and afterwards felt bad about himself/herself? |- |48 |agbi48 |HB |mix | | | | | |1 |Extremes  of Mood & Energy |Have you found that your child’s thinking changes greatly – that there are periods of several days or more when he/she thinks better than most people, and other periods when his/her mind doesn’t work well at all? |- |49 |agbi49 |dep |dep | | | | | |12 |low energy/anhedonia |Have there been times of a day or more when your child had no feelings or emotions and seemed cut off from other people? |- |51 |agbi51 |HB |hyp | | | | | |5 |high drive |Have there been times when your child has done things – like perhaps driving recklessly, taking a trip on the spur of the moment, creating a public disturbance, being more sexually active than usual, getting into fights, destroying property, or getting into trouble with the law – which he/she later thought showed poor judgment? |- |52 |agbi52 |dep |dep | |sleep | |F1 | |14 |sleep disturb (dep) |Has your child had periods of sadness and depression when, for several days or more, it took him/her over an hour to get to sleep at night, even though he/she was very tired? |- |55 |agbi55 |dep |dep | | | | | |15 |dep irritable mood |Have there been times when upsetting or bad thoughts kept going through your child’s mind and he/she couldn’t stop them? |- |57 |agbi57 |HB |hyp | | | | | |7 |racing thoughts, cog disturb (up) |Have there been times when your child had blank spells in which his/her activities were interrupted, and he/she did not know what was going on around him/her? |- |58 |agbi58 |dep |dep | | | | | |19 |atypical dep features |Has your child had sad and depressed periods of several days or more, interrupted by periods lasting between an hour to a day when he/she felt extremely happy and intensely energetic? |- |60 |agbi60 |dep |dep | | | | | |17 |somatic sx (appetite & sleep) |Has your child experienced weight changes (increases, decreases, or both) of five (5) pounds or more in short periods of time (three weeks or less), not including changes due to physical illness, menstruation, exercise, or dieting? |- |61 |agbi61 |HB |hyp | | | | | |8 |grandiosity |Have there been periods of a couple days or more when your child’s sexual feelings and thoughts were almost constant, and he/she couldn’t think about anything else? |- |65 |agbi65 |dep |dep | | | | | |17 |somatic sx (appetite & sleep) |Have there been times of several days or more when your child felt very down and depressed during the early part of the day, but then less so during the evening? |- |66 |agbi66 |HB |hyp | | | | | |4 |elated mood |Have then been times when your child began many new activities with lots of enthusiasm and then found himself/herself quickly losing interest in them? |- |67 |agbi67 |dep |dep | | | | | |17 |somatic sx (appetite & sleep) |Have you found that your child’s mood consistently follows the seasons, where he/she has long periods of depression during the winter but mostly happy periods during the summer? |- |69 |agbi69 |dep |dep | | | | | |20 |tearful, sad appearance |Have there been times of several days or more when your child has struggled to control an urge to cry, has had frequent crying spells, or found himself/herself crying without really understanding why (other than related to the menstrual cycle)? |- |70 |agbi70 |dep |dep | | | | | |11 |loss of interest |Have there been times of several days or more when almost all sexual interest was lost? |- |71 |agbi71 |dep |dep | | | | | |16 |excess guilt & paranoia |Has your child found himself/herself at times feeling fearful or suspicious or his/her environment or people around him/her? | |} ==== Reliability ==== For all versions of the GBI, the full length scales have exceptionally high internal consistency reliability. This is due to a combination of the scale length (28 or 46 items) and the items asking about related symptoms, often in blends. The length of the scales and the high reading level make them less useful in many clinical settings. Item Response Theory (IRT) provides a different way of estimating the reliability of test scores that is not tied to the length of the scale. The IRT approach also has the advantage of seeing how reliable scores are across the range of the underlying trait. For the GBI, IRT would show whether the reliability stays at acceptable levels even at low levels of depression or manic symptoms (as would often be seen if using the scale in a general community setting or screening), as well as at the high end of mood symptom severity (as might be encountered in a hospital). These figures compare the reliability for several of the short forms based on self report (i.e., teenagers using the GBI to describe themselves). {| class="wikitable" |+ Item Response Theory (IRT) estimates of reliability for self-report on the GBI !10 Item Short Forms !7 Item Short Forms |- ![[File:Reliability of the GBI Depression 10 item Mania scale, based on Item Response Theory.png|thumb]] ![[File:Reliability of the 7 Up mania form based on Item Response Theory.png|thumb]] |- |[[File:Reliability of the GBI Depression 10 item form A, based on Item Response Theory.png|thumb]] |[[File:Reliability of the 7 Down depression form based on Item Response Theory.png|thumb]] |- |[[File:Reliability of the GBI Depression 10 item form B, based on Item Response Theory.png|thumb]] | |} === The GBI or P-GBI for assessing the probability of mood disorders === The diagnostic accuracy of the test depends on the base rate of disorders for your sample. The positive and negative predictive value are directly influenced by base rate. However, the sensitivity and specificity of a test also can vary from sample to sample (Kraemer, 1992). For this reason, the cut scores published in any article cannot be assumed to be equally valid in new contexts. Please refer to the GBI Manual (available from Depue) and the monograph published in the ''Journal of Abnormal Psychology,<ref name=":1">{{Cite journal|last1=Depue|first1=Richard A.|last2=Slater|first2=J. F.|last3=Wolfstetter-Kausch|first3=H.|last4=Klein|first4=Daniel N.|last5=Goplerud|first5=E.|last6=Farr|first6=D. A.|date=1981|title=A behavioral paradigm for identifying persons at risk for bipolar depressive disorder: A conceptual framework and five validation studies|journal=Journal of Abnormal Psychology|volume=90|issue=5|pages=381–437|doi=10/dcr7nc|issn=KSL}}</ref>'' for additional information about the measure. Two meta-analyses have included the GBI, one in youths under age 18,<ref name=":2">{{Cite journal|last1=Youngstrom|first1=Eric A.|last2=Genzlinger|first2=Jacquelynne E.|last3=Egerton|first3=Gregory A.|last4=Van Meter|first4=Anna R.|date=2015|title=Multivariate meta-analysis of the discriminative validity of caregiver, youth, and teacher rating scales for pediatric bipolar disorder: Mother knows best about mania|journal=Archives of Scientific Psychology|volume=3|issue=1|pages=112–137|doi=10/gf6zrb|issn=2169-3269}}</ref> and the other as a self-report measure in adults.<ref name=":3">{{Cite journal|last1=Youngstrom|first1=E. A.|last2=Egerton|first2=G. A.|last3=Genzlinger|first3=J.|last4=Freeman|first4=L. K.|last5=Rizvi|first5=S. H.|last6=Van Meter|first6=A.|date=2018-02-01|title=Improving the global identification of bipolar spectrum disorders: Meta-analysis of the diagnostic accuracy of checklists|journal=Psychological Bulletin|volume=144|pages=315–342|doi=10/gc9fzw|issn=1939-1455 (Electronic) 0033-2909 (Linking)}}</ref> The GBI was in the top tier of measures in terms of diagnostic accuracy in both meta-analyses. However, the short forms have not had their diagnostic accuracy published in adult samples (i.e., all published work used the 73 item version).<ref>{{Cite journal|last1=Youngstrom|first1=E. A.|last2=Murray|first2=G.|last3=Johnson|first3=S. L.|last4=Findling|first4=R. L.|date=2013-12|title=The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory|journal=Psychological Assessment|volume=25|issue=4|pages=1377–83|doi=10/f5kp9c|issn=1939-134X (Electronic) 1040-3590 (Linking)}}</ref> Bipolar disorder is rare in most clinical settings (e.g., prevalence of less than 10% in outpatient and private practices, and 2-4% in the general population).<ref>{{Cite journal|last1=Merikangas|first1=Kathleen R.|last2=Pato|first2=Michael|date=2009|title=Recent developments in the epidemiology of bipolar disorder in adults and children: Magnitude, correlates, and future directions|journal=Clinical Psychology: Science and Practice|volume=16|issue=2|pages=121–133|doi=10/bdj38s|issn=1468-2850}}</ref> Because of the low “base rate,” most people scoring high on any screening test are likely to '''not''' have the condition.<ref name=":4">{{Cite book|title=Evidence-based medicine: How to practice and teach EBM|last1=Straus|first1=Sharon E.|last2=Glasziou|first2=Paul|last3=Richardson|first3=W. Scott|last4=Haynes|first4=R. Brian|date=2011|publisher=Churchill Livingstone|edition=4th|location=New York, NY}}</ref> Put another way, the “false positives” will outnumber the “true positives” in most situations unless bipolar disorder is fairly common where one is using the test. [[File:Probability nomogram -- useful for combining probability and new information that changes odds, as used in Evidence-Based Medicine and Evidence-Based Assessment 01.pdf|thumb|Nomogram for combining likelihood ratios and probabilities]] The preferred method for using these tools would be to focus on the change in likelihood of a bipolar diagnosis based on high and low scores. Low scores on a good test decrease the odds that a given youth has a bipolar disorder, just as high scores should increase the odds. It is possible to formally combine (1) the change in odds associated with a test score and (2) the prior probability that the youth had a bipolar diagnosis to obtain a new estimate of the probability that the child has bipolar disorder. This can be done visually (using a “[[w:Nomogram|nomogram]]”), mathematically,<ref>{{Cite journal|last=Webb|first=M.P.K.|last2=Sidebotham|first2=D.|date=2020|title=Bayes' formula: a powerful but counterintuitive tool for medical decision-making|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7808025/|journal=BJA Education|volume=20|issue=6|pages=208–213|doi=10.1016/j.bjae.2020.03.002|issn=2058-5349|pmc=7808025|pmid=33456952}}</ref> by use of a table containing the posterior probabilities for a fixed prevalence, or using an online calculator.<ref>{{Cite web|url=http://psych.fullerton.edu/mbirnbaum/bayes/bayescalc.htm|title=Bayesian Calculator|website=psych.fullerton.edu|access-date=2021-10-14}}</ref> There are several excellent sources for clinicians who are interested in learning more about using changes in odds as a way of refining diagnosis,<ref name=":4" /> including the [[Evidence based assessment/Prediction phase|Prediction]] section in the materials about Evidence Based Assessment on Wikiversity. The changes in odds (or diagnostic likelihood ratios) associated with scores on six different tests (the P-GBI, the P-YMRS, the Achenbach CBCL, TRF, and YSR, and the self-report GBI) based on a large sample of outpatients,<ref>{{Cite journal |last1=Youngstrom |first1=E. A. |last2=Findling |first2=R. L. |last3=Calabrese |first3=Joseph R. |last4=Gracious |first4=B. L. |last5=Demeter |first5=C. |last6=DelPorto Bedoya |first6=D. |last7=Price |first7=Megan |date=2004 |title=Comparing the diagnostic accuracy of six potential screening instruments for bipolar disorder in youths aged 5 to 17 years |journal=Journal of the American Academy of Child & Adolescent Psychiatry |volume=43 |pages=847–858 |doi=10/fqjsq6 }}</ref> and an update based a more recent review is available.<ref name=":0">{{Cite book |title=Assessment of Disorders in Childhood and Adolescence |last1=Youngstrom |first1=Eric A. |last2=Morton |first2=Emma |last3=Murray |first3=Greg |date=2020 |publisher=Guilford Press |isbn=978-1-4625-3898-0 |editor=Eric A. Youngstrom |editor2=Mitchell J. Prinstein |editor3=E. J. Mash |editor4=R. Barkley |edition=5th |location=New York, NY |chapter=Bipolar disorder }}</ref> We also are including a table here that is based on these likelihood ratios, estimating the probability that a child has bipolar disorder assuming a base rate of 5% in combination with a test score in the particular range. We chose the 5% base rate estimate for three reasons: (1) because other colleagues are estimating that 5% of the youths evaluated at outpatient academic research centers meet criteria for a bipolar spectrum disorder (e.g., 6-7% of outpatient cases evaluated in the TEAM multi-site NIMH grant; Geller et al., 2002); (2) because 5% is low enough to serve as a reminder that bipolar disorder is likely to be rare in community mental health, outpatient, and private practice settings, yet high enough to act as a reminder that the disorder can occur and should be assessed; (3) because a 5% base rate will be reduced to negligible probabilities by low or moderate scores on good tests, and raised to intermediate probabilities (30% to 50% range) by high scores on the same tests. If bipolar disorder is substantially more rare or more common at your site than 5%, we strongly recommend using a rate compared to [[Evidence based assessment/Bipolar disorder in youth (assessment portfolio)#Base rates of PBD in different clinical settings and populations|benchmarks from similar settings]] as the starting point. <br /> {| class="wikitable" |+Teenager or young adult filling out a version of GBI about themselves |Target Diagnosis: |Bipolar |Bipolar |Bipolar |Any Mood |- |Scale |MDQ |GBI-10M |7 Up |GBI-10Da |- |Shortest |12 |10 |7 |10 |- |Reading Level^a |7.3 |11.1 |11.1 |11.1 |- |Languages |13+ |25+ |4+ |25+ |- |Projected ''d'' |.40 |.43 |.36 | |- |Projected AUC (95% ''CI'') |.61 (.54 to .67) |.62 (.58 to .67) |.60 (.56 to .65) |.66 (.62 to .70) |- |Sensitivity at Specificity=.9 |.20 |.22 |.20 |.23 |- |DiLR+ |2.0 raw 9+ |2.2 raw 19+ |2.0 raw 11+ |2.3 raw 16+ |- |Time Frame |Lifetime |Past year |Past year |Past year |} ^a[[w:Flesch–Kincaid readability tests|Flesch-Kincaid]] estimate of grade level. {| class="wikitable" |+Parent or primary caregiver using GBI to describe their youth's mood symptoms |Target Diagnosis: |Bipolar |Bipolar |Bipolar |Any Mood |- |Scale |PGBI-10M |CMRS |FIRM |PGBI-10Da |- |Length (items) |10* |10* |1/2 page |10* |- |Reading Level^a |11.1 |6.5 |7.6 |11.1 |- |Languages |25+ |5 |2 |25+ |- |Projected ''d'' |1.30 |.87 |.47 |1.30 |- |Projected AUC (95% ''CI'') |.82 (.80 to .84) |.73 (.66 to .80) |.63 (.54 to .72) |.82 (.80 to .84) |- |Sensitivity at Specificity=.9 |.47 |.21 |.28 |.52 |- |DiLR+ |4.7 raw 15.5+ |2.1 raw 12+ |2.8 raw 8+ |5.2 raw 10+ |- |Time Frame |Past year |Lifetime |Lifetime (family history) |Past year |} === Using the GBI to measure treatment response === The GBI has been used in several treatment studies, and it shows good sensitivity to treatment effects. The 10 item versions in particular are brief enough to be repeated during the course of treatment, but show similar effect sizes to interview-based ratings in research studies.<ref>{{Cite journal|last=Youngstrom|first=Eric|last2=Zhao|first2=Joan|last3=Mankoski|first3=Raymond|last4=Forbes|first4=Robert A.|last5=Marcus|first5=Ronald M.|last6=Carson|first6=William|last7=McQuade|first7=Robert|last8=Findling|first8=Robert L.|date=2013-03|title=Clinical Significance of Treatment Effects with Aripiprazole versus Placebo in a Study of Manic or Mixed Episodes Associated with Pediatric Bipolar I Disorder|url=http://www.liebertpub.com/doi/10.1089/cap.2012.0024|journal=Journal of Child and Adolescent Psychopharmacology|language=en|volume=23|issue=2|pages=72–79|doi=10.1089/cap.2012.0024|issn=1044-5463|pmc=PMC3696952|pmid=23480324}}</ref> The 7 Up-7 Down scales have not been tested in an extracted, standalone format in treatment studies yet. Here are benchmarks for evaluating change during treatment:<ref name=":0" /> {| class="wikitable" | | colspan="3" |'''Benchmarks*''' | colspan="3" |'''Critical Change''' '''(Raw Scores)''' |'''[[w:Minimal important difference|Minimal Important Difference]]''' |- |'''Measure''' |''Away'' |''Back'' |''Closer'' |''95%'' |''90%'' |''SE_difference'' |''(MID)'' ''d ~.5'' |- |'''PGBI-10M^a''' |1 |9 |6 |6 |5 |3.2 |3 |- |'''PGBI-10Da^a''' |<nowiki>--</nowiki> |7 |4 |6 |5 |3.0 |3 |- |'''PGBI-10Db^a''' |<nowiki>--</nowiki> |7 |4 |6 |5 |2.9 |3 |- |'''AGBI-10M^c''' |<nowiki>--</nowiki> |14 |7 |6 |5 |3.1 |3 |- |'''AGBI-10Da^c''' |<nowiki>--</nowiki> |18 |7 |6 |5 |3.2 |3 |- |'''AGBI-10Db^c''' |<nowiki>--</nowiki> |16 |7 |6 |5 |2.9 |4 |- |'''7 Up^c''' |<nowiki>--</nowiki> |8 |4 |4 |4 |2.2 |3 |- |'''7 Down^c''' |'''--''' |'''12''' |'''5''' |'''5''' |'''4''' |'''2.3''' |3 |} <nowiki>*</nowiki>The benchmarks are based on clinical and nonclinical norms, following the "[[w:Clinical significance|clinically significant]] change" model by Jacobson and colleagues. ==== Interpretive example for measuring treatment progress and outcome ==== Juan's mother fills out a PGI-10M and PGBI-10Da as part of an evaluation. Both of these have raw scores that range from 0 to 30. Juan initially scores a 21, which is in a high risk range for potential bipolar disorder. After the feedback and first therapy session, the score comes down to a 17 (4 point drop). This is larger than the "Minimally Important Difference" ('''MID''') of 3, suggesting that this is large enough for the person to believe that treatment might be helping some. However, the amount of change needed to be be confident that treatment is actually contributing to reliable change would need to be larger: The '''95% confidence in change''' target is 6 points for this measure (equating to a '''reliable change index''' > 1.96 in Jacobson's approach). After several months of treatment, Juan's score according to his mother's report is down to a 7. This is enough to be confident that treatment is helping. The 14 point reduction (21-7 = 14 point difference) not only exceeds the targets for MID and reliable change, but it also is lower than the "'''Back'''" into the normal range threshold of 9. The Back threshold is the 95th percentile for a reference group without bipolar disorder (in this case often with other mild or moderate clinical issues, as there is no nonclinical standardization sample for the PGBI, like most clinical symptom assessments). Scores this high are likely to still be noticeable and may be concerning to others, but they are also within the range of what could also occur for other reasons besides having a bipolar disorder, including problems in daily living as a youth or adult. The Back threshold is the most liberal of the "clinically significant change" definitions proposed by Jacobson and colleagues. Reducing the score to a 6 or lower would satisfy Jacobson's "'''Closer'''" definition -- reliable change combined with a score more typical of the nonbipolar than bipolar reference groups (operationally defined as the weighted mean of the two groups). Again, scores of 5-6 may be noticeable and sometimes irritating, but they also are a marked improvement compared to where Juan started. This would be an even more impressive example of clinically significant change. If treatment continued and succeeded in getting his score down to a 0 or 1, that would not only show near complete elimination of the symptoms, but it also would satisfy Jacobson's most stringent definition of clinically significant change -- getting the score '''Away''' from the clinical reference group (e.g., below the 2.5th percentile of the clinical reference group). This is an exceptionally stringent definition, and impossible to achieve with many outcome measures, where two SDs below average would require negative raw scores. == Evidence Base == === Peer Reviewed Research === The first paper published on the GBI was in 1981,<ref name=":1" /> and research has appeared steadily since then. The GBI consistently has exceptional evidence of reliability, due to its combination of length and well-written (but complex) items. It has showed excellent evidence of discriminative validity in two meta-analyses, one focused on self-report in adults<ref name=":3" /> and the other looking at performance with children and teenagers.<ref name=":2" /> Miller et al. (2009) noted, in their review of assessment instruments for adult bipolar spectrum disorders: <blockquote>As a diagnostic screening tool, the scale with the best support is the GBI, as it has consistently demonstrated sensitivity of approximately .75 and specificity above .97. Readers should be cautious, however, because multiple versions of the scale exist, and cutoffs for a positive screen have not been firmly established.<ref>Christopher J. Miller, Sheri L. Johnson, and Lori Eisner, "Assessment Tools for Adult Bipolar Disorder," ''Clinical Psychology: Science and Practice'' 16, no. 2 (2009): 192–193. <nowiki>https://doi.org/10.1111/j.1468-2850.2009.01158.x</nowiki></ref></blockquote> '''''PubMed Search:''''' Click [https://www.ncbi.nlm.nih.gov/pubmed/?term=%22General+Behavior+Inventory%22 here] for a current search on PubMed, a free database that covers medicine (so some articles published in psychology journals might be missing). The entries will usually include abstracts, and sometimes will include a version of full text (especially if the project was grant funded). The search is designed to be highly specific (i.e., not including lots of irrelevant articles), but it might miss some articles. == Languages Available == The GBI has been translated into multiple languages. Some of the different languages available are linked here. There is a repository that includes many of these here. An older version of this subset is hosted on Trello [https://trello.com/b/dYUKlNRP/translated-measures-dashboard here]. If you are looking here, note that there are separate columns for the 7 Up-7 Down, General Behavior Inventory (self-report), and parent report versions. == Research Resources == === Supplemental Materials === Two papers that tested several short forms when used as parent report<ref name=":5">{{Cite journal|last=Youngstrom|first=Eric A.|last2=Meter|first2=Anna Van|last3=Frazier|first3=Thomas W.|last4=Youngstrom|first4=Jennifer Kogos|last5=Findling|first5=Robert L.|date=2018-07-24|title=Developing and Validating Short Forms of the Parent General Behavior Inventory Mania and Depression Scales for Rating Youth Mood Symptoms|url=https://doi.org/10.1080/15374416.2018.1491006|journal=Journal of Clinical Child & Adolescent Psychology|volume=0|issue=0|pages=1–16|doi=10.1080/15374416.2018.1491006|issn=1537-4416|pmid=30040496}}</ref> and as adolescent self report included supplemental materials that provided more detail about methods and results. These supplemental materials are published here so that they are freely accessible and archived (rather than having them only behind a publisher's paywall). ==== Factor Structure of the Short Forms ==== Tables ==== Rationale for the Ranking of Expected Criterion Correlations ==== Both papers had two samples, an academic clinic and a community mental health center, along with a large set of variables that could be used to examine the criterion validity of the short forms compared to the full length GBI scales.<ref name=":5" /> Here is the detailed description of how the authors ranked the criterion correlations from what they expected to be largest to smallest: The GBI scales were expected to show the highest correlation with the cognate rating scale on the Youth Self Report (YSR) because they were converging measures of the same trait, they were completed by the same informant (i.e., they shared method variance)(Podsakoff, MacKenzie, & Podsakoff, 2012), and they were continuous scales (not categorical variables, which would shrink the size of the observed correlation even when measuring the same construct) (Cohen, 1988). The YSR Internalizing score was expected to show the highest correlation because of the shared method variance: both it and the GBI were completed by the same person. They would be expected to correlate ''r'' ~.3 to .4 even if they measured different constructs, due to response set, mood congruent biases, and other factors unrelated to the trait (Podsakoff et al., 2012). Further, Internalizing and Externalizing correlate ''r'' ~.6 in the standardization sample (Achenbach & Rescorla, 2001)<ref name=":6">Achenbach, T.M., & Rescorla, L.A. (2001). Manual for the ASEBA School-Age Forms & Profiles. Burlington, VT: University of Vermont, Research Center for Children, Youth, & Families.</ref> and also in our samples. Finally, the 28-item and 10-item GBI versions included some “mixed” items, and so they had depression content embedded in them. The 7 Up, in contrast, was “purer” and showed lower correlations with Internalizing in both samples (though still > .4). The YSR Externalizing score was the best available converging measure for the mania scales in the two samples, but it was not expected to show quite as high criterion correlations as the depression-Internalizing coefficients. A meta-analysis (Youngstrom, Genzlinger, Egerton, & Van Meter, 2015) of diagnostic accuracy shows that Externalizing is not as strongly associated with bipolar disorder as the GBI is: The effect size was ''r'' ~.45 for parent ratings on measures like the GBI, versus ''r'' ~.34 for measures such as the CBCL Externalizing; ''r'' ~.26 for GBI versus ''r'' ~ .13 for YSR correlations with diagnoses. The Externalizing score does not include items asking about grandiosity, inflated self-esteem, elevated or expansive mood, or decreased need for sleep without fatigue – the “handle” symptoms that are more specific to hypomania and mania (Craney & Geller, 2003; Youngstrom, Birmaher, & Findling, 2008). Put simply, Externalizing is not as good a measure of the mania construct as the GBI scales are, so the criterion correlation with it is not going to be stellar. Next, the youth and parent correlations use different sources, eliminating the shared method variance component. Meta-analyses find that parent-youth agreement about the same trait in the youth hovers in the ''r'' ~.2 to .3 range (Achenbach, McConaughy, & Howell, 1987; De Los Reyes et al., 2015), exactly what we see in the Academic sample and similar to the estimates in the Community sample. For the correlations with the diagnoses and interview-based severity ratings: Meta-analyses have established that parent report is significantly more strongly related to youth diagnoses than youth self-report is (Stockings et al., 2015; Youngstrom et al., 2015). Converting the effect sizes from Youngstrom et al. 2015 into correlations yields an estimate of ''r'' ~ .45 for parent ratings and corresponding youth diagnoses, versus .26 for youth ratings and their own diagnoses. The same pattern will hold for the YMRS and CDRS-R as the diagnoses – they were based on the same interview as the KSADS diagnoses, and so they correlate with the diagnosis ''r'' > .9. Because of attenuation artifacts when using a categorical variable (i.e., diagnosis) instead of a continuous one (i.e., severity on the YMRS or CDRS-R), we would expect the correlations with diagnosis to be about 80% of the size of the correlation with the severity rating (Cohen, 1988). Depression scores were expected to show a small to moderate correlation with age as well as with female sex based on normative data (e.g., patterns in Internalizing scores in the standardization sample for the ASEBA; Achenbach & Rescorla, 2001).<ref name=":6" /> Anxiety diagnoses were expected to show a small to moderate correlation with depression scales due to overlapping symptoms (e.g., the tripartite model of depression and anxiety) (Chorpita & Daleiden, 2002; Watson, Clark, et al., 1995; Watson, Weber, et al., 1995). Last in the rankings were some demographic variables (e.g., race) and unrelated diagnoses that were expected to have near-zero correlation coefficients. == External Links == * The Open Translations Project ([https://trello.com/b/dYUKlNRP/translated-measures-dashboard TOpTraP]) -- an effort to gather the translated versions of the best free measures in one place. The GBI 10 item mania and depression scales are available in more than two dozen languages; the full length version is available in several. *[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/severe-mood-swings-and-bipolar-spectrum-disorders/ EffectiveChildTherapy.Org information on Bipolar Disorder] -- a website built for families to learn more about ways to improve social, emotional, and academic life for youths *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] -- the professional society for psychologists focusing on helping youths and families dealing with emotional and behavioral challenges ==Share this page== '''Share this page by clicking the following social media and interactive platforms: ''' <small>{{Share email|icon=true|subject=Journal article: {{SUBPAGENAME}}|contents={{{email|http://en.wikiversity.org/wiki/{{FULLPAGENAMEE}}}}}}} | {{Share facebook|icon=true}} | {{Share 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text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{Navbox | state = uncollapsed | name = Free and validated psychological instruments scoring information | title = Free and validated psychological instruments scoring information | listclass = hlist | navbar = plain | group1 = Anxiety disorders | list1 = {{Navbox|child | group1 = Anxiety | list1 = [[{{BASEPAGENAME}}/Screen for child anxiety related disorders|Screen for Child and Anxiety Related Disorders]]{{•}} [[{{BASEPAGENAME}}/Beck Anxiety Inventory|Beck Anxiety Inventory]] | group2 = Obsessive-compulsive and related disorders | list2 = [[{{BASEPAGENAME}}/Yale-Brown Obsessive Compulsive Scale|Yale–Brown Obsessive Compulsive Scale]] | group3 = Phobias | list3 = | group4 = Tic disorders | list4 = [[{{BASEPAGENAME}}/Yale Global Tic Severity Scale|Yale Global Tic Severity Scale]] }} | group2 = Autism spectrum disorders | list2 = [[{{BASEPAGENAME}}/Autism Treatment Evaluation Checklist|Autism Treatment Evaluation Checklist]] | group3 = ADHD | list3 = [[{{BASEPAGENAME}}/Vanderbilt ADHD Diagnostic Rating Scale|Vanderbilt ADHD Diagnostic Rating Scale]]{{•}} [[{{BASEPAGENAME}}/ADHD Rating Scale|ADHD Rating Scale]] | group4 = Behavior problems | list4 = [[{{BASEPAGENAME}}/Modified Overt Aggression Scale|Modified Overt Aggression Scale]]{{•}} [[{{BASEPAGENAME}}/School refusal assessment scale-revised|School Refusal Assessment Scale- Revised]] | group5 = Eating disorders | list5 = [[{{BASEPAGENAME}}/Eating Attitudes Test|Eating Attitudes Test]] | group6 = Mood disorders | list6 = [[{{BASEPAGENAME}}/Center for Epidemiological Studies Depression Scale|Center for Epidemiological Studies Depression Scale]]{{•}} [[{{BASEPAGENAME}}/Child Mania Rating Scale|Child Mania Rating Scale]]{{•}} [[{{BASEPAGENAME}}/General Behavior Inventory|General Behavior Inventory]]{{•}} [[{{BASEPAGENAME}}/Hypomania Checklist|Hypomania Checklist]]{{•}} [[{{BASEPAGENAME}}/Mood Disorder Questionnaire|Mood Disorders Questionnaire]]{{•}} [[{{BASEPAGENAME}}/Mood and Feelings Questionnaire|Mood and Feelings Questionnaire]]{{•}} [[{{BASEPAGENAME}}/Patient Health Questionnaire|Patient Health Questionnaire]]{{•}} [[{{BASEPAGENAME}}/Weinberg screen affective scale|Weinberg Screen Affective Scale]]{{•}} [[{{BASEPAGENAME}}/Young Mania Rating Scale|Young Mania Rating Scale]] | group7 = Posttraumatic stress disorder | list7 = [[{{BASEPAGENAME}}/Child PTSD symptom scale|Child PTSD Symptom Scale]]{{•}} [[{{BASEPAGENAME}}/UCLA post-traumatic stress disorder reaction index|UCLA PTSD Index for DSM-IV]] | group8 = Suicide | list8 = [[{{BASEPAGENAME}}/Columbia Suicide Severity Rating Scale|Columbia Suicide Severity Rating Scale]]{{•}} [[{{BASEPAGENAME}}/Suicide behavior questionnaire-revised|Suicide Behaviors Questionnaire-Revised]] | group9 = Sleep | list9 = [[{{BASEPAGENAME}}/Children Sleep Habits Questionnaire|Children Sleep Habits Questionnaire]]{{•}} [[{{BASEPAGENAME}}/Pittsburgh Sleep Quality Index|Pittsburgh Sleep Quality Index]] | group10 = Substance use disorders | list10 = [[{{BASEPAGENAME}}/Alcohol Use Disorders Identification Test|Alcohol Use Disorders Identification Test]]{{•}} [[{{BASEPAGENAME}}/CAGE questionnaire|CAGE Questionnaire]]{{•}} [[{{BASEPAGENAME}}/CRAFFT Screening Test|CRAFFT]]}} 24r3btfo2qn7otux6n6claznmkuqygh 0 221175 2691667 2476380 2024-12-12T17:39:44Z DavidMCEddy 218607 /* 2.11. US foreign interventions in opposition to democracy */ 2691667 wikitext text/x-wiki {{Essay}} :''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].'' * '''''Those whom the gods wish to destroy they first make mad.'''''<ref>Anonymous ancient proverb, wrongly attributed to Euripides. {{Citation | title = Euripides | publisher = Wikiquote | url = https://en.wikiquote.org/wiki/Euripides | accessdate = 2017-02-19}}</ref> This essay (''a'') reviews evidence suggesting that the [[w:War on Terror|War on Terror]] is not going well, (''b'') surveys research that provides a credible explanation for why it’s not going well, and (''c'') recommends minimizing the use of force and focusing instead on rule of law and on subsidizing democratically managed media to manage armed conflicts including terrorism and the Islamic State. [[File:Terrorism deaths worldwide.svg|thumb|Figure 1. Terrorism deaths worldwide, 1970-2015.<ref name=Graves2017a>using data from the Global Terrorism Database (GTD), summarized by Graves (2017).</ref>]] Terrorist activity worldwide has grown dramatically since 2012, at least according to terrorism deaths recorded in the [[w:Global Terrorism Database|Global Terrorism Database (GTD)]] summarized in Figure 1.<ref>Graves (2017). Questions have been raised about the quality of GTD data, especially its consistency over time and whether individual events are or are not classified as suicide terrorism. For suicide terrorism in particular, the GTD is not consistent with the [[w:Suicide Attack Database|Suicide Attack Database]] maintained by the [[w:Chicago Project on Security and Terrorism|Chicago Project on Security and Threats (CPOST)]]. We used the GTD, because it seems to be the best data available on the subject, and we don’t believe its defects raise substantive questions about our conclusions.</ref> In the following, we (1) note that terrorism is minuscule as a cause of death nearly everywhere, (2) review the literature on the long-term impact of alternative responses to terrorism and conflict more generally, (3) discuss the role of the media in shaping public reactions to terrorism (and virtually any other public policy issue), and (4) summarize implications of the above for personal action and public policy. == 1. Terrorism is minuscule as a cause of death == Before discussing possible contributors to the recent spike in terrorism deaths, we first note that terrorism is essentially minuscule as a cause of death, except for a small number of countries with active armed conflicts: Even in the worst year on record, 2014, terrorism deaths were less than 0.08 percent (one twelfth of one percent or 800 per million) of all deaths worldwide that year. More generally, terrorism has been responsible for the deaths of 0.02 percent (one fiftieth of one percent or 200 per million) of all the people who have died since the first entry in the Global Terrorism Database in 1970.<ref>This assumes that the death rate numbers from the World Bank are adequate for present purposes; see Graves (2017). Data published by the CIA often differ from the World Bank. We have not explored those discrepancies but would be surprised if the differences raise substantive questions about our conclusions.</ref> [[w:List of causes of death by rate|You're several times more likely to die from a fire (0.55%) than terrorism. Or from accidental poisoning (0.61%). Or drown (0.67%). Or die from a fall (0.69%, using World Health Organization data from 2002).]] Certainly, deaths are not the only problem from terrorism: Terrorist attacks also injure people , and destroy property. However, analysis of the numbers of incidents and people injured essentially tell the same story. The research summarized in this essay suggests that this recent spike in terrorism is a ''product'' of the militarization of conflicts (section 2 below) driven by a [[w:Fascination with death|fascination with death]] and how that interacts with media funding and governance (section 3 below). * ''The primary problem from terrorism seems to be the collateral damage from military responses used to combat it.'' Unfortunately, collateral damage was not mentioned in either the index or the table of contents of ''An end to evil: How to win the War on Terror'' by [[w:David Frum|Frum]] and [[w:Richard Perle|Perle]], which appeared in January 2004; that was roughly nine months after the [[w:2003 invasion of Iraq|US-led invasion of Iraq in 2003]] and seven after [[w:Mission Accomplished speech|President Bush's "Mission Accomplished" speech]]. Instead, they complained, "Pessimism and defeatism have provided the sound track to the war on terrorism from the beginning".<ref>Frum and Perle (2004, p. 8)</ref> The evidence summarized in this essay suggests that "pessimism and defeatism" have played far less of a role than collateral damage in the ensuing violence that has engulfed that region since 2003; this includes the rise of ISIL and the recent spike in terrorism documented in Figures 1 and 2 and in Appendix 1. We next consider terrorism in France, the US, and the dozen countries most impacted by this recent spike in terrorism before reviewing research relating to these recommendations. === 1.1. Terrorism in France and the United States === [[File:Terrorism deaths in France.svg|thumb|Figure 2. Terrorism deaths in France. The spike in 2015 is over 6 times the previous maximum since 1970 and is indicated by a number off the scale.<ref name=Graves2017a/>]] Figure 2 plots terrorism deaths in [[France]] through 2015. The number for 2015 is labeled, not plotted, because it is over 6 times the second largest recorded number of terrorism deaths in France since the first entry in the [[w:Global Terrorism Database|Global Terrorism Database]] (GTD) in 1970, and plotting it would make it difficult to see the earlier variability. GTD data for 2016 are not yet available. However, the Wikipedia [[w:List of terrorist incidents in France|"list of terrorist incidents in France"]] reports 89 deaths for that year. That’s just over half the number for 2015 and over three times the previous maximum.<ref> {{Citation | title = List of terrorist incidents in France | publisher = Wikipedia | url = https://en.wikipedia.org/wiki/List_of_terrorist_incidents_in_France | accessdate = 2017-02-17}}</ref> These relatively high numbers have made security a key issue in the French presidential campaign in progress as this is being written.<ref>{{Citation | last = Giudicelli | first = Anne | date = 2017-02-15 | title = Elections in France: It’s all about security | journal = Al Jazeera | url = http://www.aljazeera.com/indepth/opinion/2017/02/elections-france-security-170215090123247.html | accessdate = 2017-02-17}}</ref> These numbers are, nevertheless, tiny as a cause of death. The 161 terrorism deaths in France in 2015 is roughly 0.03 percent (one thirtieth of a percent or 300 per million) of all French deaths that year.<ref>This assumes that the death rate in France is roughly 8.4 per thousand population, which is the number for 2014 (the most recent available) in the World Bank WDI.xlsx data, and the population of France is roughly 67 million. {{Citation | date = December 2016 | title = World Development Indicators - Downloads: WDI (Excel)-ZIP (80 MB) | publisher = World Bank | url = http://data.worldbank.org/data-catalog/world-development-indicators | accessdate = 2017-03-11}}</ref> [[File:Terrorism deaths in the United States.svg|thumb|Figure 3. Terrorism deaths in the United States. The spike in 2001 is labeled, not plotted, because it is almost 20 times the death toll from the second largest terrorist attack in US history, the [[w:Oklahoma City bombing|Oklahoma City bombing]], which killed 168 people in 1995.<ref name=Graves2017a/>]] The two biggest terrorist attacks on [[w:United States|US]] soil were the [[w:September 11 attacks|September 11, 2001 attacks]] that took roughly 3,000 lives, and the [[w:Oklahoma City bombing|Oklahoma City bombing]] that killed 168 people in 1995.<ref>{{Citation | title = Oklahoma City bombing | publisher = wikipedia | url = https://en.wikipedia.org/wiki/Oklahoma_City_bombing | accessdate = 2017-02-17}}</ref> For the US, [[w:Global Terrorism Database|GTD]] records show no recent spike comparable to that for the world and France in Figures 1 and 2; see Figure 3. The most recent years suggest a modest upward trend -- possibly a return to the environment of the early 1970s, but nothing like 2001 nor the recent worldwide or French numbers. The GTD records 1,397 US citizens killed in terrorist incidents between 1970 and 2000, and 943 between 2002 and 2015,<ref>The GTD records 2,910 US citizens killed in terrorist incidents in 2001. The official number of people killed in the [[w:September 11 attacks|September 11 attacks]] was 2,996. The difference is non-US citizens killed in those attacks.</ref> for an average of 116 per year over these 46 years; without 2001, it averages only 52 per year. For 2001 through 2016, an average of 257 US citizens were killed per year. Averaged over the 46 years in the Global Terrorism Database, terrorism has taken the lives of 0.005 percent (one half of one hundredth of one percent or 50 per million) of all Americans who died during that period.<ref name=Graves2017>Graves (2017)</ref> To put these numbers into perspective, we provide three comparisons: * [[w:List of motor vehicle deaths in U.S. by year|42,196 people were killed on US highways in 2001, averaging 3,516 per month]].<ref> {{Citation | title = List of motor vehicle deaths in U.S. by year | publisher = wikipedia | url = https://en.wikipedia.org/wiki/List_of_motor_vehicle_deaths_in_U.S._by_year | accessdate = 2017-03-07}}</ref> Thus, more people were killed in the average month in 2001 than in the worst terrorist incident ever recorded. Between 2001 and 2015, 569,229 people died on US highways and 3,939 died from terrorist attacks -- a ratio of 145 to 1. * Roughly [[w:male breast cancer|440 ''men'' die in the US each year due to breast cancer.]] (This doesn't count breast cancer among women, who are roughly 100 times as likely to get it as men.)<ref>Females are more likely than males to survive breast cancer, because their cancers are usually caught earlier. Male breast cancers are not caught earlier, because the risks are so low that it has so far never seemed worth the effort to develop sensible screening procedures for it. {{Citation | title = Male breast cancer | publisher = wikipedia | url = https://en.wikipedia.org/wiki/Male_breast_cancer | accessdate = 2017-02-26}}</ref> That’s roughly 0.03 percent (one thirtieth of a percent or 300 per million) of all male deaths in the US. Thus, breast cancer has taken the lives of roughly six times as many men in the US as terrorism since the first entry in the Global Terrorism Database (GTD) in 1970. This rate, 0.03 percent, is the same rate as ''the worst year on record for France'', and roughly a third of the recent worldwide spike in Figure 1. * Between 1999 and 2003, a total of 1,676 Americans were reported to have drowned in a bathtub, hot tub or spa, averaging 335 a year.<ref>{{cite Q |Q60226981 }}</ref> In other words, America’s highways and tubs are greater risks than terrorism, and breast cancer is a greater risk even for males, except in countries with active armed hostilities like Iraq. Beyond this, as noted above, you are several times more likely to die from a fire, accidental poisoning, drowning or a fall. This is not to trivialize terrorist deaths, but only to say that we should not spend more money on protection against terrorism than the threat deserves -- and we should avoid actions that could make it worse, as suggested by the evidence summarized here. === 1.2. Countries with the most terrorism deaths 2014-2015 === [[File:Terrorism deaths by country, 2014-2015.svg|thumb|Figure 4. Terrorism deaths by country, 2014-2015, per the Global Terrorism Database.<ref name=Graves2017/>]] Figure 4 summarizes the total number of terrorism deaths by country in 2014 and 2015. France and the US are buried in the thirteenth “other” category in this plot. Terrorism is not a substantive problem for France or the US or anywhere else except for the relatively small number of countries with active armed hostilities, identified in Figure 4.<ref name=Graves2017a/> Not one of these countries had a comparable terrorism problem prior to the announcement of the US-led “[[w:War on terror|War on Terror]]”. This is clear from plots similar to Figures 1-3 for each of these dozen countries individually (available in Appendix 1). Pakistan, Egypt, and Sudan had terrorism problems prior to 2001 but nothing comparable to what they’ve experienced since the US declared a War on Terror. This claim is supported by more than just the relatively tiny number of deaths and injuries. It is also supported by research on the long-term impact of alternative approaches to conflict. This is called here "[[effective defense]]" and summarized in the next section. == 2. Research on the long-term impact of alternative approaches to conflict == * When people are killed and property destroyed, the apparent perpetrators often make enemies.<ref name=Graves2004>Graves (2004)</ref> * [[w:David Petraeus|General David Petraeus]] as commander of US Central Command understood that “you can't kill your way out of an insurgency, … [Y]ou have to find other kinds of ammunition, and it's not always a bullet," according to one of his closest colleagues.<ref>{{cite news | last1 = Depaulo | first1 = Lisa | title = Leader of the Year: Right Man, Right Time | url=http://www.gq.com/story/leader-of-the-year-general-david-petraeus-war | accessdate = 2017-02-19 | publisher = GQ | date = October 31, 2008 }}</ref> * [[w:Stanley McChrystal|General Stanley McChrystal]], who held several command positions in Iraq and Afghanistan, wrote, "we found that nearly every first-time jihadist claimed [that the torture at] Abu Ghraib had first jolted him into action." He also said that, "mistreating detainees would discredit us. ... The pictures [from] Abu Ghraib represented a setback for America's efforts in Iraq. Simultaneously undermining US domestic confidence in the way in which America was operating, and creating or reinforcing negative perceptions worldwide of American values, it fueled violence".<ref><!-- Stanley McChrystal (2013) My share of the task: A memoir-->{{cite Q|Q72893267}}</ref> The research reviewed here suggests that the world would be safer, more prosperous, and more democratic if the West treated terrorism as a law enforcement issue, strengthening international law, while dramatically reducing its reliance on military force. We need more research to better understand what drives people off the sidelines to support one side or the other in conflict and what motivates them to increase or decrease their level of support and to defect. === 2.1. How terrorist groups end === [[File:RANDterroristGpsEnd2006.svg|thumb|Figure 5. How terrorist groups end (''n'' = 268): The most common ending for a terrorist group is to convert to nonviolence via negotiations (43 percent), with most of the rest terminated by law enforcement (40 percent). Groups that were ended by military force constituted only 7 percent.<ref>Jones and Libicki (2008, p. 19)</ref>]] In 2008 two researchers with the [[w:RAND Corporation|RAND Corporation]], [[w:Seth Jones|Seth Jones]] and [[w:Martin C. Libicki|Martin Libicki]], discussed all the terrorist groups they could find that were active between 1968 and 2006: they found 648. Of those, 136 splintered, 244 were still active, leaving 268 that had ended. Of the ones that ended, 83 percent succumbed to rule of law, including 43 percent converting to non-violent political actors and 40 percent taken out by law enforcement. Only 20 groups, 7 percent, were defeated by military action; 10 percent won.<ref>Jones and Libicki (2008). For detailed analysis of a few cases, see {{Citation | last = Cronin | first = Audrey Kurth | year = 2009 | title = How terrorism ends: Understanding the decline and demise of terrorist campaigns | publisher = Princeton U. Pr. | isbn = 978-0-691-15239-4}}</ref> When Jones and Libicki focused only on terrorist groups that became large enough to be called an “insurgency,” like the [[w:Islamic State of Iraq and the Levant|Islamic State]], the percentages changed: 18 of 38 (47 percent) were ended by negotiations. 10 (26 percent) ended in victory for the insurgents. 8 (21 percent) succumbed to military force. 2 (5 percent) were suppressed by law enforcement.<ref>Jones and Libicki (2008, p. 101, Table 5.1)</ref> Thus, when a terrorist group converted to an insurgency, the use of military force increased. Perhaps most importantly, the effectiveness of law enforcement fell dramatically at the expense of major increases in victories by both the terrorists and the military.<ref>The percentage of cases ending in negotiated settlements increased slightly.</ref> Jones and Libicki concluded by recommending “that United States should make police and intelligence efforts the backbone of US counterterrorism policy and move away from its mantra of fighting a war on terrorism.”<ref>Jones and Libicki (2008, p. 8)</ref> Using data from Jones and Libicki (2008), Bapat found that US military aid has tended to ''reduce'' the incentives of recipient governments to negotiate, thereby ''prolonging'' the threat.<ref>Bapat (2011)</ref> In other words, to the extent that Bapat's analysis is accurate, the War on Terror has been more a war ''for'' terror than ''against'' terror. * ''Why is the West using the least effective approach to terrorism (the military)?'' * ''To what extent are Western governments pressuring other countries to respond militarily to terrorism rather than relying on law enforcement and negotiations, as Bapat claims? Are the results really as negative as Bapat suggests?'' === 2.2. The long-term impact of alternative approaches to conflict === Chenoweth and Stephan (2011) identified all the major governmental change efforts of the twentieth century.<ref>Their database includes all violent and nonviolent campaigns ending between 1900 and 2006 that involved over 1,000 people at some point with a goal of changing the government.</ref> They found 217 movements that were predominantly violent and 106 that were primarily nonviolent. Outcomes were classified as either (1) failure, (2) partial success or (3) success. The basic results are summarized in Table 1: Nonviolence was twice as likely to succeed as violence. {|class="wikitable" |+ '''Table 1'''. Major governmental change efforts of the twentieth century by dominant nature of the struggle (violent or nonviolent) and by outcome (failure, partial success, success) in the NAVCO1.1 data set compiled by Chenoweth and Stephan.<ref>{{Citation | last = Chenoweth | first = Erica | author-link = w:Erica Chenoweth | year = 2011 | title = Nonviolent and Violent Campaigns and Outcomes (NAVCO) Dataset, v. 1.1 | publisher = University of Denver | url = http://www.du.edu/korbel/sie/research/chenow_navco_data.html | accessdate = 2014-10-08}}</ref> |- ! !colspan="2"|{{center top}}'''Number of conflicts'''{{center bottom}} !colspan="2"|{{center top}}'''Percent'''^(*){{center bottom}} |-Primary nature -> ! !!violent!!nonviolent!!violent!!nonviolent |- |'''Outcome''' || || || || |- | align="right"|success||align="right"|55||align="right"|57||align="right"|25%||align="right"|54% |- | align="right"|partial success||align="right"|28||align="right"|26||align="right"|13%||align="right"|25% |- | align="right"|failure||align="right"|134||align="right"|23||align="right"|62%||align="right"|22% |- | align="right"|total||align="right"|217||align="right"|106||align="right"|100%||align="right"|100%^(*) |- | colspan="5" align="left"|^(*) Percent within conflicts of the same primary nature. Thus, the "violent" column percents add to 100. The nonviolent total differs from 100 only because of round-off. |} There has been some study of whether the existence of a {{w|radical flank effect|radical flank}} increases or decreases the likelihood of success of a primarily nonviolent movement. Chenoweth and Schock (2015) said that, "no study has systematically evaluated the effects of simultaneous armed resistance on the success rates of unarmed resistance campaigns." To fill this gap, they studied which of the 106 primarily nonviolent campaigns in Chenoweth and Stephan (2011) had a radical flank. They concluded that, "large-scale maximalist nonviolent campaigns often succeed despite intra- or extramovement violent flanks, but seldom because of them.”<ref>{{cite Q|Q83970885}}<!-- Do Contemporaneous Armed Challenges Affect the Outcomes of Mass Nonviolent Campaigns?, in refereed journal ''Mobilization'' -->. Other studies cited in the Wikipedia article on "[[w:radical flank effect|Radical flank effect]] reached the opposite conclusion; however, these other studies were earlier and smaller. In addition, they seemed to be less systematic than Chenoweth and Schock.</ref> However, the benefits of nonviolence extend beyond the end of a conflict. Chenoweth and Stephan merged their data with the [[w:Polity data series|Polity IV database]], which 'is a widely used data series [summarizing] annual information on the level of democracy for all independent states with greater than 500,000 total population and covers the years 1800–2013. ... For each year and country, a "Polity Score" is determined, which ranges from -10 to +10, with -10 to -6 corresponding to [[w:Autocracy|autocracies]], -5 to 5 corresponding to [[w:Anocracy|anocracies]], and 6 to 10 to [[w:Democracy|democracies]].'<ref>{{Citation | title = Polity data series | publisher = Wikipedia | url = https://en.wikipedia.org/wiki/Polity_data_series | accessdate = 2017-02-26}}</ref> {|class="wikitable" |+ '''Table 2'''. Polity Score ranges from -10 to +10 |- ! !!minimum value!!maximum value |- |align="right"|[[w:Autocracy|autocracies]]||-10||-6 |- |align="right"|[[w:Anocracy|anocracies]]||-5||5 |- |align="right"|[[w:Democracy|democracies]]||6||10 |- |} Table 3 shows the average increase in democratization from one year before the start of a conflict to one, five, and ten years after. The results suggest that ''win or lose'', nonviolence tends on average to be followed by an increase in the Polity IV rating while violence has relatively little impact on democratization. As noted above, nonviolence builds democracy, while violence perpetuates tyranny, on average, in the long run. {|class="wikitable" |+ '''Table 3'''. Average increase in Polity score from one year before to 1, 5 and 10 years after a conflict.^(*) |- | || colspan="3" style="text-align: center;" | violent | colspan="3" style="text-align: center;" | nonviolent |- ! years after !! 1 !! 5 !! 10 !! 1 !! 5 !! 10 |- |align="right"|success||0.5||-1.6||-.5||5.9||10.1||10.0 |- |align="right"|partial success||1.4||2.1||1.9||4.2||6.8||7.6 |- |align="right"|failure||0.4||0.8||0.8||3.0||2.7||4.9 |- | statistically significant || colspan="3" style="text-align: center;" | no | colspan="3" style="text-align: center;" | yes |- | colspan="7" align="left"|^(*) None of the changes following violent campaigns are statistically significant while all the changes following nonviolent campaigns are significant at the 0.05 level, and all but three have significance probabilities less than 0.001. |} The reality is more complicated than the simple summary of Table 3: A primary determinant of the level of democracy after a conflict, apart from the primary (violent or nonviolent) nature of the conflict, is the level of democracy before. Figure 6 plots the Polity IV democracy score 5 years after each conflict vs. 1 year before.<ref>Plots of data from the NAVCO1.1 database; see Chenoweth and Stephan (2011).</ref> [[File:Democratization 5 years after vs. 1 year before twentieth century revolutions.svg|thumb|Figure 6. Democratization 5 years after (vertical scale) vs. 1 year before (horizontal scale) twentieth century revolutions]] This plot includes six panels grouped by the primary nature (violent or nonviolent) of the conflict and the outcome (failure, partial success, success). Points on the dotted diagonal line in each panel indicate conflicts that were accompanied by zero change in their Polity scores for the indicated time frame. The solid lines in each panel are based on the best fit of several models considered. This expresses the democracy score after the conflict as linearly dependent on the democracy score before plus interactions between outcome and both the democracy score before and the primary nature of the conflict.<ref>For more detail, see Chenoweth and Stephan (2011).</ref> These plots show more detail behind the simple summary of Table 3: Successful nonviolent revolutions have on average had a substantial impact in increasing the level of democracy among autocracies but no impact among the best democracies. By contrast, the worst long term outcomes tend to be from ''successful violent'' revolutions. This is worth repeating: * ''Successful violent revolutions provide the worst prospects for democracy in most cases.'' This can be explained by observing that successful violence brings to power people who know how to use violence but are not as good at solving problems without violence. (The comparable analyses of democratization 1 and 10 years after the end of the conflict are essentially the same; those plots are in Appendix 2.) In sum, the overall image supports the claim made above: Win or lose, nonviolence builds democracy, while violence perpetuates tyranny, on average, in the long run. This is consistent with the findings of Jones and Libicki (2008) mentioned above, that better outcomes for democracy are achieved when governmental officials support rule of law and negotiations. More on this comes from research on why people obey the law, when they do. === 2.3. Why people obey the law === People tend to obey the law, when they do, when [[w:Procedural justice|legal procedures seem fair to them]]. [[w:Tom R. Tyler|Tyler and Huo (2002)]] concluded that people of different ethnicities in the US have essentially the same concept of justice as majority whites but different experiences. This was based on a survey of African-Americans, Hispanics, and whites. They describe two alternative strategies for effective law enforcement: * Deterrence: effective but inefficient * Process-based: efficient and effective Tyler and Huo's analysis suggests that biased, unprofessional behavior of police, prosecutors and judges not only produces concerns of injustice, it cripples law enforcement efforts by making it more difficult for police and prosecutors to obtain the evidence needed to convict guilty parties. This is important for winning the War on Terror, because it describes some of the negative consequences of official behaviors that convince substantive segments of society that law enforcement is unjust. Retired US Green Beret Lt. Col. D. Scott Mann described how US Special Forces units defeated the Taliban by living for extended periods in small Afghan villages, treating the local people with respect, and showing sensitivity to their culture and concerns. In this way, they gradually earned people's trust to the point that people would inform them of Taliban activities in that geographic area and other problems. Where this was not done, Mann said the Taliban was winning.<ref>{{cite Q|Q83934350}}<!--Game Changers --></ref> For more extreme cases, we turn to the work of [[w:Robert Pape|Robert Pape]] on “dying to win” and “bombing to win”. === 2.4. Dying to win === Suicide terrorism is a primary focus of the [[w:Chicago Project on Security and Terrorism|Chicago Project on Security and Terrorism]] founded by [[w:Robert Pape|Robert Pape]] at the [[w:University of Chicago|University of Chicago]]. These data were discussed in his (2005) ''[[w:Dying to Win: The Strategic Logic of Suicide Terrorism|Dying to Win: The Strategic Logic of Suicide Terrorism]]'' and his (2010) ''Cutting the Fuse: The Explosion of Global Suicide Terrorism and How to Stop It'', with James K. Feldman. ''Dying to Win'' analyzed 315 suicide terrorism attacks around the world from 1980 through 2003. ''Cutting the Fuse'' evaluated more than 2,100 attacks, over 6 times the number in the first book. Pape and Feldman’s conclusions include the following: * "Overall, foreign military occupation accounts for 98.5% -- and the deployment of American combat forces for 92% -- of all the 1,833 suicide terrorist attacks around the world in the past six years [2004-2009]."<ref>Pape and Feldman (2010, p. 28)</ref> * "Have these actions ... made America safe? In a narrow sense, America is safer. There has not been another attack on the scale of 9/11. ... In a broader sense, however, America is not safer. Anti-American suicide terrorism is rapidly rising around the world."<ref>Pape and Feldman (2010, p. 318)</ref> * "[I]n both Iraq and Afghanistan ... local communities that did not inherently share the terrorists' political, social, and military agenda, eventually support[ed] the terrorists organization's campaign ... after local communities began to perceive the Western forces as an occupier ... as foreign troops propping up and controlling their national government, changing their local culture, jeopardizing economic well-being, and conducting combat operations with high collateral damage ... . But, we have also seen in Iraq that this perception of occupation can be changed ... ."<ref>Pape and Feldman (2010, p. 333)</ref> * "For over a decade our enemies have been dying to win. By ending the perception that the United States and its allies are occupiers, we can cut the fuse to the suicide terrorism threat."<ref>Pape and Feldman (2010, p. 335)</ref> The terrorist attacks of September 11, 2001, fit Pape’s model: The US maintained a substantive military presence in Saudi Arabia from the [[w:Gulf War|1990-91 Persian Gulf War]] until 2003. [[w:United States withdrawal from Saudi Arabia|It seems virtually certain that without those foreign troops on Saudi soil, Al Qaeda could not have found 19 young men willing to commit suicide on September 11, 2001, to send a message to the people of the United States.]]<ref>The US has rejected the characterization of its presence as an "occupation", noting that the government of Saudi Arabia consented to the presence of troops. However, the dominant factor in the motivation of our opposition is how ‘’they’’ perceive it, not how the US perceives it.</ref> This position is supported by US government documents declassified on July 15, 2016, that reported that some of the suicide mass murderers of September 11, 2001, had received help from employees of the Saudi Embassy and Consulates in the US, including members of the Saudi royal family, to obtain housing and other things they needed to get the training required to do what they did on that fateful day. Moreover, ranking officials in the [[w:Presidency of George W. Bush|George W. Bush administration]] knew of this complicity at least in 2002 before the US-led invasion of Iraq and successfully convinced the [[w:Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001|Joint congressional inquiry into the terrorist attacks of September 11, 2001]] to redact [[w:The 28 Pages|28 pages]] containing that information from their December 2002 report.<ref>See references cited in the Wikipedia article on [[w:The 28 Pages|”The 28 Pages”]].</ref> * ''Given the documented support of Saudi government officials for the September 11 attacks, why did the US invade Afghanistan and Iraq?'' * ''We have enemies, because we have friends like these.'' It seems worth noting in this context that negotiations for the [[w:Turkmenistan–Afghanistan–Pakistan–India Pipeline|Turkmenistan–Afghanistan–Pakistan–India Pipeline]] were suspended after the [[w:1998 United States embassy bombings|1998 United States embassy bombings]] over Taliban support for [[w:Osama bin Laden|bin Laden]], who had been accused of masterminding those 1998 bombings. Pipeline construction began in 2015 without the need for a jury trial of bin Laden. This coincidence does not prove that the pipeline was part of the motivation for invading Afghanistan after September 11, 2001, but the coincidence is striking. === 2.5. Bombing to win === ''Bombing to win'' by [[w:Robert Pape|Robert Pape]] provides a qualitative survey of all the uses of [[w:Airpower|airpower]] up to the early 1990s. He concluded that ''strategic bombing was wasteful,'' and the only uses of airpower that contributed to military victory involved support of ground operations. He made a possible exception for nuclear weapons, but noted that when the Japanese Emperor [[w:Hirohito|Hirohito]] informed his military of the need to surrender after the [[w:Atomic bombings of Hiroshima and Nagasaki|atomic bombings of Hiroshima and Nagasaki]], he did NOT mention the atomic bomb. Instead he noted the Soviet entry into the war and the rapid collapse of the elite Japanese forces in Manchuria that followed. Hirohito’s true motives are unclear, because when he spoke to the civilian population, he mentioned the atomic bombings and not the Soviets.<ref>Pape (1996) </ref> Regarding nuclear weapons, retired US Army Colonel [[w:Andrew Bacevich|Andrew Bacevich]] claimed that, “Nuclear weapons are unusable. Their employment in any conceivable scenario would be a political and moral catastrophe. … [T]hey are unlikely to dissuade the adversaries most likely to employ such weapons against us -- Islamic extremists … . If anything, the opposite is true. By retaining a strategic arsenal in readiness ..., the United States continues tacitly to sustain the view that nuclear weapons play a legitimate role in international politics”.<ref>{{Citation | last = Bacevich | first = Andrew J. | year = 2008 | title = The Limits of Power: The End of American Exceptionalism | publisher = Metropolitan Books | pages = 178-179 | isbn = 0805090169}}</ref> Bacevich’s concern is strengthened by an estimate of the probability distribution of the "[[time to extinction of civilization]]". That analysis concludes that there is a probability of between 10 and 20 percent of a nuclear war in the next 40 years that would loft so much soot into the stratosphere where rain clouds rarely form and where most of it would remain for decades preventing up to 70 percent of the sunlight from reaching the surface of the earth. This would produce a nuclear winter during which roughly 98 percent of humanity would starve to death. Considerations like these have driven some senior US statesmen like [[w:Sam Nunn|former US Senator Sam Nunn]], [[w:William Perry|former US Secretary of Defense William Perry]], [[w:Henry Kissinger|former US Secretary of States Henry Kissinger]] and [[w:George Shultz|George Shultz]] to support [[w:nuclear disarmament|nuclear disarmament]], though not necessarily unilaterally. On the broader question of the effectiveness of strategic bombing, the [[w:United States Air Force|US Air Force (USAF)]] funded a 1999 study by the [[w:RAND Corporation|RAND corporation]] on that. It did NOT support Pape’s conclusions.<ref>{{Citation | last =Byman | first =Daniel L. | author-link = w:Daniel Byman | last2 =Waxman | first2 =Matthew C. | author2-link =w:Matthew Waxman | last3 = Larson | first3=Eric | title = Air Power as a Coercive Instrument | publisher =RAND Corporation | series =Project AIR FORCE | year =1999 | url = https://www.rand.org/content/dam/rand/pubs/monograph_reports/2007/MR1061.pdf | accessdate = July 29, 2013}}</ref> However, this RAND study produced a database of “all instances of [[w:Airpower|air power]] coercion from 1917 to 1999”, which was used by Horowitz and Reiter (2001). Applying multivariate probit analysis, Horowitz and Reiter concluded the following: # Coercion is more effective when the target's military vulnerability is higher. # Higher levels of civilian vulnerability have no effect on the chances of coercion success. # Target regime type ([[w:Polity data series|its Polity score]]) has no effect. # Success is less likely when the attacker demands the target change its leadership.<ref>Horowitz and Reiter (2001)</ref> The first two of these four conclusions provide a more solid empirical basis for Pape’s claims. The fourth of these conclusions seems consistent with the observation of Jones and Libicki (2008) discussed above, that 43 percent of terrorist groups ended with a negotiated political settlement. One interpretation of this is that the collateral damage from strategic bombing can easily be viewed as excessive by the victims and people on the sidelines. However, with ground operations, it’s generally less obvious who should be blamed for collateral damage. As noted above, we need more research to better understand what drives people off the sidelines to support one side or the other in conflict and what motivates them to increase or decrease their level of support and to defect. Moreover, this research should be funded and managed by sources independent of anyone with a [[w:conflict of interest|conflict of interest]] in the conclusions. This information should be compiled in more or less in real time with at most a few months delay in making it available to the public. The research available today suggests that the impact of collateral damage on the evolution of conflict may be substantially greater than is acknowledged by those driving current US military operations. As noted in the introduction to this section, when people are killed and property destroyed, the apparent perpetrators often make enemies.<ref name=Graves2004/> However, until the public gains a better understanding the counterproductive nature of collateral damage, we cannot expect wisdom in this area to weigh very heavily in the selection of political and military leaders. A tragic example is how the G. W. Bush administration politicized intelligence to justify invading Iraq. People were fired for insisting that the available evidence did not support claims that Saddam Hussein had weapons of mass destruction or links to al Qaeda.<ref>Goodman (2013, ch. Four. Bush's Surrender to the Pentagon)</ref> [[Effective defense|We need more research on how to win (or better avoid) wars]] -- and more public awareness of this distinction. Without this, it's easy for the side with a bigger military to win battles and lose wars. Generals and admirals are too often pushed to choose strategies that resonate with their superiors but manufacture enemies faster than they can be neutralized. This is part of how the British lost the American Revolution<ref>{{Citation | last = Graves | first = Spencer B. | year = 2005 | title = Violence, Nonviolence and the American Revolution | url = http://prodsyse.com/conflict/Nonviolence&AmericanRevolution.pdf | accessdate = 2015-11-26}}</ref> and how the US lost its war in Vietnam,<ref>Chayes (2015)</ref> to name only two examples. There is a growing body of evidence that “collateral damage” is rarely neutral. === 2.6. Drones === The Obama administration has claimed that drones ([[w:Unmanned combat aerial vehicle|unmanned combat aerial vehicles]]) are over 95 percent effective in killing enemy combatants.{{Citation needed | date=March 2017}} Their opposition, including many former drone pilots, who have spoken out at risk of being prosecuted for exposing classified information, claim that the figure may be closer to 50 percent.{{Citation needed | date=March 2017}} The discrepancy is explained by the claim that people killed in a drone strike are classified as EKIA (enemy killed in action) unless there is substantial evidence to the contrary.<ref>{{Citation needed | date=March 2017}} We need further study of the impact on Pakistani politics of US drone operations there and the contribution of US drone operations in Yemen to the civil war in progress as this is being written, 2017-03-06.</ref> === 2.7. Islamic terrorism === There is evidence to suggest that the two most effective recruiters for Islamic terrorism may be Saudi Arabia and the United States. * As noted in the discussion above of "dying to win", without US troops on Saudi soil 1991-2001, no Islamic terrorist organization would likely have found 19 young men willing to commit suicide in a terrorist attack on September 11, 2001. * On June 5, 2002, then-FBI Director {{w|Robert Mueller}} said "that investigators believe the idea of the Sept. 11 attacks on the World Trade Center and Pentagon came from al Qaeda leaders in Afghanistan, the actual plotting was done in Germany, and the financing came through the United Arab Emirates from sources in Afghanistan."<ref>{{cite Q|Q61755017}}<!-- Mueller outlines origin, funding of Sept. 11 plot -->.</ref> However, [[w:The 28 pages|"The 28 Pages"]], declassified 2016-07-15 by then-President {{w|Barack Obama}} document that at least as early as 1999, the FBI had information of Saudi government funding and support for what later became the [[w:September 11 attacks|suicide mass murders of September 11, 2001]]. Moreover, when Mueller made those comments, the FBI was actively working to prevent the {{w|Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001}} from obtaining that information. [[w:Bob Graham|Bob Graham]], who served on that Joint Inquiry, said in 2015 that during that inquiry and since, the FBI went "beyond just covering up ... into ... aggressive deception."<ref>{{cite Q|Q65002265}}<!-- Florida Ex-Senator Pursues Claims of Saudi Ties to Sept. 11 Attacks -->.</ref> In that 2015 comment, Senator Graham said the FBI "had investigated a Saudi family in Sarasota, Fla., and had found multiple contacts between it and the hijackers training nearby until the family fled just before the attacks." By the time of that report, Graham also doubtless knew of the allegations by [[w:Sibel Edmonds|Sibel Edmonds]] that in the spring and summer of 2001, months before the September 11 attacks, an Iranian ex-patriot had told the FBI that bin Laden's network was planning to use airplanes in terror attacks in four or five major cities including New York City, Chicago, Washington DC, and San Francisco; possibly Los Angeles or Las Vegas. After September 11, Edmonds' co-workers got "an absolute order [that they] never got any warnings." Graham doubtless also knew of "a classified military planning effort led by the U.S. Special Operations Command (SOCOM) and the Defense Intelligence Agency (DIA)" called [[w:Able Danger|Able Danger]], which "had identified two of three al-Qaeda cells active in the 9/11 attacks". Government official denied having had knowledge of this before the September 11 attacks, but Lt. Col. [[w:Anthony Shaffer (intelligence officer)|Anthony Shaffer]] claims he had such information from his work on Able Danger before the attacks and was ordered not to testify to these congressional committees. * Those "28 pages" include a discussion of an {{w|America West}} flight that made an emergency landing in 1999 when two men tried to break into the cockpit. The apparent perpetrators showed the FBI Saudi passports and tickets apparently paid by the Saudi embassy in Washington, DC. That information was classified "Top Secret", presumably to keep it from raising too many questions about Saudi complicity in the suicide mass murders of September 11, 2001. * By 2002, the G. W. Bush administration had already invaded Afghanistan ostensibly to capture Osama bin Laden after refusing to consider the offer of the Afghani government to extradite him if the US provided evidence of his culpability. And Bush administration officials were working hard to convince the public in the US and their "coalition of the willing" to support invading Iraq, which they did in March 2003. * The growth of the [[w:Islamic State of Iraq and the Levant|Islamic State of Iraq and the Levant]] (ISIL), also called the "Islamic State of Iraq and Syria" (ISIS) or "Daesh", appears to have been a product of both (a) excessive collateral damage and (b) media censorship (discussed below) that enabled corruption to grow excessively in the post-Saddam Iraqi government and military. [[w:Islamic State of Iraq and the Levant|A former Chief Strategist in the Office of the Coordinator for Counterterrorism]] of the US State Department, [[w:David Kilcullen|David Kilcullen]], said that "There undeniably would be no Isis if we had not invaded Iraq."<ref name="Indie ISIL">{{citation |url=http://www.independent.co.uk/news/world/middle-east/iraq-war-invasion-caused-isis-islamic-state-daesh-saysus-military-adviser-david-kilcullen-a6912236.html |title=Former US military adviser David Kilcullen says there would be no Isis without Iraq invasion |work=The Independent |date=4 March 2016 |accessdate=8 March 2016 }}</ref> [[w:Graham Fuller|Graham Fuller]], a former CIA agent, was quoted in a 2014 interview as follows: "I think the United States is one of the key creators of [[w:Islamic State of Iraq and the Levant|[ISIS]]]. The United States did not plan the formation of ISIS, but its destructive interventions in the Middle East and the war in Iraq were the basic causes of the birth of ISIS."<ref>{{Citation | last = Basaran | first = Ergi | last2 = Fuller | first2 = Graham | date = 2014-09-02 | title = Former CIA officer says US policies helped create IS | journal = Al-Monitor: The pulse of the Middle East | url = https://www.al-monitor.com/pulse/politics/2014/09/turkey-usa-iraq-syria-isis-fuller.html | accessdate = 2017-12-05}}</ref> * Other sources note that, “[A]lmost all of ISIL's leaders ... are former Iraqi military and intelligence officers, … who lost their jobs and pensions in the [[w:De-Ba'athification|de-Ba'athification]] process” undertaken by the US-led occupation. “ISIL is a [[w:theocracy|theocracy]], [[w:proto-state|proto-state]] and a [[w:Salafi movement|Salafi]] or [[w:Wahhabism|Wahhabi]] group.”<ref>{{Citation | title = Islamic State of Iraq and the Levant | publisher = wikipedia | url = https://en.wikipedia.org/wiki/Islamic_State_of_Iraq_and_the_Levant | accessdate = 2017-03-04}}</ref> * The [[w:Salafi movement|Salafi]] / [[w:Wahhabi|Wahhabi]] branch of Islam is the most violent form of Islam, promoted by the Saudi royal family in part by funding schools and mosques throughout the Muslim world that taught their branch of Islam. At the same time, {{w|Structural adjustment}} programs pushed by the US through the {{w|World Bank}} and the {{w|International Monetary Fund}} have allegedly made life more difficult for all but the ultra-wealthy in many poor countries while pushing those countries to reduce their funding for education. As a result, the Saudi-funded {{w|Wahhabism|Wahhabi}}-{{w|Salafi movement|Salafist}} schools became the primary educational alternative for the children of many poor people. Many ISIL fighters reportedly came out of such schools. * ISIL relies mostly on captured weapons. For example, in Mosul between 4 and 10 June 2014 a group of between 500 and 600 ISIL troops “were able to seize six divisions’ worth of strategic weaponry, all of it US-supplied” from a force with a paper strength of 120,000 men.<ref name='AlJ5'><!-- Enemy of Enemies: The Rise of ISIL. Chapter 5. 2009-2015: Syria uprising and ISIL in Syria-->{{cite Q|Q113710863}}</ref> This invasion included [[w:Robert Pape|suicide attacks, which are almost always motivated by a foreign occupation (as discussed above)]].<ref>Pape (2005), Pape and Feldman (2010)</ref> [[w:Fall of Mosul|The invading troops]] were faced by an army where “every officer had to pay for his post”, and made money from soldiers who would kick back “half their salaries to their officers in return for staying at home or doing another job”, and from receiving funds to feed an organization three times the size on paper as were actually there.<ref>{{Citation | last = Astore | first = William J. | date = 2014-10-14 | title = Tomgram: William Astore, America's Hollow Foreign Legions -- Investing in Junk Armies | publisher = TomDispatch.com | url = http://www.tomdispatch.com/post/175907/tomgram%3A_william_astore,_america%27s_hollow_foreign_legions/ | accessdate = 2014-10-16}}</ref><ref>{{Citation | year = 2015 | title = Enemy of Enemies: The Rise of ISIL | chapter = 2. 2004-2006: Abu Musab Al-Zarqawi Emerges | publisher = Al Jazeera | url = http://interactive.aljazeera.com/aje/2015/riseofisil/chapter-two.html | accessdate = 2015-11-27}}</ref><ref>Beyond this, there have been numerous allegations that [[w:Finances of ISIL|Saudi Arabia and Qatar may have clandestinely provided funds to ISIL]], though that has not been proven.</ref> * In 2008 {{w|Stuart A. Levey}}, the Under Secretary for Terrorism and Financial Intelligence in the US Department of the Treasury, told the US Senate Finance Committee that “Saudi Arabia today remains the location where more money is going to terrorism, to Sunni terror groups and to the Taliban than any other place in the world."<ref>{{cite Q |Q61889276 }}</ref> In October 2010 he reported “significant improvement in the partnership between the U.S. and Saudi Arabia in targeting al Qaeda financing.”<ref>{{cite Q |Q61890613}}. See also [[w: Stuart A. Levey#Under Secretary for Terrorism and Financial Intelligence]]</ref> However, two months later, Wikileaks published leaked US diplomatic cable saying that, “Private individuals in Saudi Arabia and other Gulf states friendly to the United States are the chief source of funding for al-Qaeda, the Taliban and other terrorist groups,” quoting a 2009 cable from US Secretary of State Hillary Clinton as saying, "It has been an ongoing challenge to persuade Saudi officials to treat terrorist financing emanating from Saudi Arabia as a strategic priority”.<ref>{{cite Q |Q61890660 }}</ref> And in August 2018 the Associated Press reported that Saudi Arabia was paying al Qaeda to help them fight the rebels in Yemen.<ref>{{cite Q |Q61890713 }}</ref> The sources cited above suggests that key decisions in the rise of Islamic terrorism were made by virtually every US president dating back to Franklin Roosevelt, but especially George W. Bush. *''We have enemies, because we have friends like these.'' *''The widespread demonization of terrorists with minimal context is an obstacle to effective action that could address the issues that drive people to support violence and escalate a conflict.''<ref>[[w:Jihad|Jihad]] “is an Arabic word which literally means striving or struggling, especially with a praiseworthy aim.” It is occasionally used to mean “Holy war,” but that is relatively rare. Thus, it is similar to the German word [[:de:w:Kampf|Kampf]], which means “struggle,” as that word is commonly used in English. The title of the infamous [[w:Mein Kampf|Mein Kampf]] by [[w:Adolf Hitler|Adolf Hitler]] simply means, “My struggle.”</ref> Fact check: Radical Islamic terrorists represent between 0.03 and 0.14 percent (between one out of 700 and one out of 3,000) of the [[w:Islam|more than 1.7 billion Muslims in the world today]], according to Brian Steed, a Lt. Col. on the faculty of the US Army Command and General Staff College in Leavenworth<ref>Steed is fluent in Arabic and has spent substantial time working in different Arabic-speaking countries. These comments were his personal, professional opinion and were not official policy of the United States government. {{Citation | last = Steed | first = Brian L. | editor-last = Harritt | editor-first = Ira | date = 2016-05-19 | title = Confronting Extremist Violence, the Refugee Crisis, and Fear: Faith Responses | chapter = Undersanding ISIS: Maneuver in the Narrative Space | chapter-url = https://www.youtube.com/watch?v=oU74V1i29tU | publisher = American Friends Service Committee | url = https://www.youtube.com/playlist?list=PLuZ74OBEmVKD7VG2UkDSR55zXYnJyztHs | accessdate = 2017-03-04 | quote = I'm speaking as a private citizen and not as a representative of the US government. ... Violent jihadists represent between 0.03% and 0.14% of Islam. (0:43 and 10:50 of 14:23 mm:ss)}}</ref> and author of a recent book on the Islamic State.<ref>{{Citation | last = Steed | first = Brian L. | year = 2016 | title = ISIS : an introduction and guide to the Islamic State | publisher = ABC-CLIO | isbn = 1440849862}}</ref> One more point about Islamic terrorist groups: Part of their appeal is their claim that the West hates Islam.{{Citation needed|date=March 2017}} It's easy to believe that if you listen to the [[w: Xenophobia|xenophobic]] rhetoric coming from people like [[w:Marine Le Pen|Marine Le Pen]] in France, and [[w:Donald Trump|Donald Trump]] in the US, both of whose electoral success is based in part on anti-immigrant, anti-Muslim platforms. Their ability to attract support would be reduced if the West increased its support for refugees, including Muslims. An important precedent for this discussion is the implicit support for the Nazis provided by the refusal of other countries to accept Jewish refugees in the late 1930s and the story behind the book and movie, [[w:Voyage of the Damned|''Voyage of the Damned:'']] The [[w:MS St. Louis|MS ''St. Louis'']] left Hamburg for Cuba on May 13, 1939. Most of her 937 passengers were Jewish refugees fleeing Nazi persecution. En route, Cuba revoked their entry visas. Only 29, most with valid visas to other countries, were allowed to enter Cuba. Of the rest, 288 eventually settled in Britain. Most of the rest died during the war, primarily in [[w:Nazi concentration camps|Nazi death camps]]. If the US, Britain, and other countries had accepted at least the vast majority of people wanting to leave Nazi Germany, it would have weakened the Nazi program in at least two ways: * The death camps would not have been built, because the Nazis would not have had enough support from their own people to prevent the refugees from leaving.<ref>{{Citation | last = Jacques | first = Jacques | year = 1993 | title = Unarmed against Hitler: Civil resistance in Europe, 1939-1943 | publisher = Praeger | isbn = 0-275-93960-X}}</ref> * It would have raised questions about the Nazi rhetoric that the [[w:Themes in Nazi propaganda|jews were parasites like tape worms or lice]]. This in turn could have made it [[w:Economy of Nazi Germany|more difficult for them to get public support for war]], possibly even reducing the strength of their military. The war likely still would have occurred, though that's not certain. If it did, the refugees would have had greater motivation to fight than just about anyone else in the countries allied against the Axis.<ref>If the US had had an open immigration policy like this, the Nazis may have sent some saboteurs posing as refugees. However, many if not all of these could have been identified by procedures that checked personal connections with other refugees: Potential saboteurs would not likely have had as many personal connections to other refugees and to people already in host countries. No screening system is perfect. However, it seems likely that the benefits to the hosts from receiving the refugees would outweigh the risks.</ref> === 2.8. The cost of US wars in the Middle East === In a campaign rally October 26, 2016, then-candidate Trump 'repeated his call to "drain the swamp," knocking the "failed elites in Washington" for being wrong about everything from foreign policy to health care. "The people opposing us are the same people — and think of this — who’ve wasted $6 trillion on wars in the Middle East — we could have rebuilt our country twice — that have produced only more terrorism, more death, and more suffering – imagine if that money had been spent at home". [[w:PolitiFact|PolitiFact]] rated Trump's $6 trillion figure as "half true", because "he is confusing money that’s been spent with money that researchers say will be spent", like obligations for benefits for combat-related disabilities of veterans that will eventually be paid over the coming decades. In this analysis, PolitiFact compared this $6 trillion number with sources giving numbers ranging from $4.8 to $7.9 trillion.<ref>{{Citation | last = Qiu | first = Linda | date = October 27, 2016 | title = Did U.S. spend $6 trillion in Middle East wars? | publisher = Politifact | url = http://www.politifact.com/truth-o-meter/statements/2016/oct/27/donald-trump/did-us-spend-6-trillion-middle-east-wars/ | accessdate = 2017-12-05}}</ref> We first note that Trump's concern that US "wars in the Middle East ... have produced only more terrorism" is consistent with the evidence summarized elsewhere in this article. However, we also wish to focus on this $6 trillion figure: That relates to wars between 2001 and 2016 -- 15 years. Thus, the US has been spending (or incurring obligations for future spending) at the rate of roughly $0.4 trillion.<ref>6/15 = 0.4</ref> By comparison, we note that the [[w:United States|US Gross Domestic Product (GDP)]] was estimated at $18.6 trillion for 2016. With this base, $0.4 trillion is over 2 percent of GDP.<ref>In 2002, US GDP was $11 trillion; $0.4 trillion was 3.6 percent of that. US GDP has been growing since then, except for a minor correction in 2009. This means that this $0.4 trillion has been declining as a percent of GDP but was always greater than 2 percent.</ref> We will return to this in the section below on "media funding and governance" and especially "citizen-directed subsidies". === 2.9. Trump and refugees === [[w:Immigration policy of Donald Trump|Very early in his Presidency, Donald Trump took several actions to implement some of the anti-immigrant policies]] that had formed a key part of his campaign. However, his justification for those policies have generally been contradicted by the available evidence. For example, he promised to suspend immigration from "areas of the world when there is a proven history of terrorism against the United States”.<ref>{{Citation | last = Qiu | first = Linda | date = June 13, 2016 | title = Wrong: Donald Trump says there's 'no system to vet' refugees | periodical = Politifact | publisher = Politifact.com | url = http://www.politifact.com/truth-o-meter/statements/2016/jun/13/donald-trump/wrong-donald-trump-says-theres-no-system-vet-refug/ | accessdate = 2017-03-10}}</ref> However, his executives orders 13769 from January 27, 2017, and 13780 from March 6, 2017, reportedly did not include any Muslim-majority countries with which he has business relations. In particular, Saudi Arabia, Egypt, Turkey, and Indonesia were not mentioned in his executive orders, even though nationals from Saudi Arabia and Egypt were directly involved in the September 11 attacks<ref>{{Citation | last =Helderman | first = Rosalind S. | date = January 28, 2017 | title = Countries where Trump does business are not hit by new travel restrictions | newspaper = Washington Post | url = https://www.washingtonpost.com/politics/countries-where-trump-does-business-are-not-hit-by-new-travel-restrictions/2017/01/28/dd40535a-e56b-11e6-a453-19ec4b3d09ba_story.html | accessdate = 2017-03-10}}</ref> -- and there is substantial documentation of other connections between Saudi Arabia and Islamic terrorism, as noted above. How were the leaders (including their staffs) who made these key decisions selected? === 2.10. Leaders and experts === [[File:Daniel KAHNEMAN.jpg|thumb|Daniel Kahneman, research psychologist who won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]] for seminal research that showed that the standard economic models of a [[w:Rational choice theory|“rational person”]] do not correspond with how humans actually think.]] Leaders and experts in many fields make worse predictions than simple rules of thumb developed by intelligent lay people, according to research psychologist [[w:Daniel Kahneman|Daniel Kahneman]]. After studying the quality of expert opinion, Kahneman concluded that “true skill” requires two things: # An environment that is sufficiently regular to be predictable. # Opportunities to learn through prolonged practice. Some fields have these attributes; others do not.<ref>Kahneman (2011, ch. 22. Expert intuition: When can we trust it?, esp. p. 240). One of Kahneman's examples is anesthesiology (p. 242), because problems with anesthetics can lead fairly quickly to death of the patient. A contrasting medical example is provided by back surgeons, discussed by Harvard Medical School Prof. [[w:Jerome Groopman|Jerome Groopman]]. He wrote that back surgery practices in the US are grandfathered to procedures used in the nineteenth century. This lack of research has retarded the development of improved procedures in that field and increased the misery of people everywhere with back pain. This research deficit is at least partly a result of lobbying the US congress by companies that manufacture devices implanted in people's backs -- and the fact that serious coverage of lobbying would threaten the profitability of the mainstream media; see the discussion of the media in this essay. {{Citation | last = Groopman | first = Jerome | author-link = w:Jerome Groopman | year = 2007 | title = How Doctors Think | publisher = Houghton Mifflin | isbn = 9780618610037}}</ref> One of Kahneman's examples involves financial markets. Two things happen every trading day. First, the financial markets either go up or down. Second, the nightly news features a pundit, who tells us why. The value of this commentary for predicting the future is zero. That's because this situation lacks sufficient regularity to support learning (Kahneman's first condition), as enough people with enough money are already in the market trying to predict it. The daily movements in prices reflect what's left and is essentially random.<ref>For a summary of the research on financial markets, see, e.g., {{Citation | last = Siegel | first = Jeremy J. | author-link = w:Jeremy Siegel | year = 2008 | title = Stocks for the Long Run, 4th ed. | publisher = McGraw-Hil | isbn = 9780071494700}}.</ref> However, claims of random variability will not attract an audience, but “experts” spouting nonsense will -- as long as the audience doesn't know it's nonsense. Kahneman's two conditions rarely apply in politics. With media primarily focused on selling behavior change in their audience to funders (as noted in the discussion of the media below), [[w:Xenophobia|xenophobic]] politicians are too often promoted while people trying to facilitate understanding and deescalation over escalation in conflict may be vilified as naive appeasers. [[w:Dick Cheney|Former Vice President Cheney]]'s [[w:One Percent Doctrine|''One Percent Doctrine'']] was used to justify torture and preventive war in the absence of substantive evidence to support it,<ref>Parton (2014)</ref> with no apparent consideration of how such policies might manufacture support for the opposition. In a survey of empirical research on "Interventions / Uses of force short of war," Prins wrote, "hawkish leaders frequently rise to power by exploiting fears of conflict escalation. The increasingly coercive polices designed to check a rival only exacerbate security concerns and deepen national perceptions of enmity."<ref>{{Citation | last = Prins | first = Brandon C. | editor-last = Denemark | editor-first = Robert A. | year = 2010 | title = The International Studies Encyclopedia | chapter = Interventions / Uses of force short of war | publisher = Blackwell Reference Online | isbn = 9781444336597 | url = http://www.isacompendium.com/public/tocnode?id=g9781444336597_yr2015_chunk_g978144433659711_ss1-57 | accessdate = 2017-03-31}}</ref> From at least some perspectives, US Vice President Dick Cheney and Israeli Prime Minister [[w:Ariel Sharon|Ariel Sharon]] might fit this description by Prins. Similar questions have been raised about how military officers are promoted.<ref>{{Citation | last = Ricks | first = Thomas E. | author-link = w:Thomas E. Ricks (journalist) | year = 2012 | title = The Generals | publisher = Penguin | isbn = 978-1-59420-404-3}}</ref> During active hostilities, the ability to win military battles can weigh heavily in promotion criteria. However, the impact of those battles on the long term outcome of the conflict is rarely a consideration. * ''It is difficult if not impossible for military and political leaders to acquire Kahneman’s “true skill,” because the long-term impact of their actions is difficult to discern in the short term and not rewarded by the current political climate.'' This short-term information deficit increases the need to collect, analyze, and disseminate information on what motivates one’s opposition. Moreover, the system for collecting and disseminating such information should be independent of the policy makers, because the temptation to suppress bad news is generally too great to resist. * ''Virtually every party to conflict thinks they know more than they do about what motivates their opposition.'' This follows from the overconfidence that virtually everyone has in the value of current knowledge, discussed below. Independent collection and dissemination of information on the motivations of people in conflict may help open paths to dialog and resolution, thereby reducing the duration and lethality of conflict. The quotes from Generals Petraeus and McChrystal in the introduction to this section on “effective defense” suggest that some leaders may understand this, at least at some level. However, these kinds of observations generally get too little coverage in the mainstream media, perhaps because they do not support the responses apparently favored by major advertisers like the major oil companies.<ref name=Focke>Focke, Niessen-Reunzi and Ruenzi (2016)</ref> (See also the discussion of media ownership, funding and profitability, below.) === 2.11. US foreign interventions in opposition to democracy === * ''Are the US and the rest of the world better off as a result of its numerous interventions in foreign countries in opposition to democracy?'' Consider the military coups that destroyed democracy in Iran 1953, Guatemala 1954, Brazil 1964, and Chile 1973: In all these cases there is solid documentation of US involvement. Beyond this, there are substantial claims that the US clandestinely supported the [[w:March 1949 Syrian coup d'état|1949 Syrian coup]]; at minimum, the [[w:Trans-Arabian Pipeline|Trans-Arabian Pipeline]], which had been held up in the Syrian parliament, was approved roughly 6 weeks after the coup. Consider also the 1952 Cuban elections, which were canceled by a military coup on March 10 organized by [[w:Fulgencio Batista|Fulgencio Batista]]. Batista had been supported by the US as de facto head of state of Cuba since 1933, but polls showed him losing badly. The US officially deplored the coup but recognized the new Batista government on March 27.<ref>{{citation | last1 = Acheson | first1 = Dean | title = Continuation of Diplomatic Relations with Cuba | url = https://history.state.gov/historicaldocuments/frus1952-54v04/d327 | website = Office of the Historian of the United States Department of State | publisher=United States Department of State | accessdate = 2017-03-09 | date = 1952-03-24 }}</ref> [[w:Fidel Castro|Fidel Castro]], a 25-year old attorney, had been running for a seat in the Cuban House of Representatives in that election. If democracy had not been overthrown in Cuba, Fidel likely would have had a career as a politician and attorney in a democratic Cuba. Similarly, [[w:Che Guevara|Che Guevara]] had been working in Guatemala at the time of the [[w:1954 Guatemalan coup d'état|1954 Guatemalan coup d'état]]. That coup turned him into a revolutionary. Without that coup, Guevara would likely have had a successful career improving public health and democracy in Latin America. Also, former President [[w:Dwight D. Eisenhower|Eisenhower]] said, "I have never [communicated] with a person knowledgeable in Indochinese affairs who did not agree that had elections been held as of the time of the fighting [[w:Battle of Dien Bien Phu|[leading to the defeat of the French in 1954]]], possibly 80 per cent of the population would have voted for the Communist Ho Chi Minh".<ref>p. 372, ch. 14. Chaos in Indochina in {{cite Q|Q61945939}}<!-- Mandate for Change -->.</ref> *''This was the universal expert consensus that was not even mentionable in the mainstream media of that day.'' {{w|Joseph McCarthy}} was elected to the US Senate in part by railing against "twenty years of treason" by the Democrats in their alleged insufficient hostility to Communism.<ref> {{cite book|last = Herman |first = Arthur |title = Joseph McCarthy: Reexamining the Life and Legacy of America's Most Hated Senator |publisher = Free Press |year = 2000 |page = [https://archive.org/details/josephmccarthyre00herm/page/131 131] |isbn = 0-684-83625-4 |url = https://archive.org/details/josephmccarthyre00herm/page/131 }}</ref> This included allegations that the Democrats had "lost China to Communism." He implied that {{w|George Marshall}}, former General and Chairman of the Joint Chiefs of Staff during World War II and Secretary of State under President Truman, of being guilty of treason.<ref name="Retreat"> {{cite book |last = McCarthy |first = Joseph |title = Major Speeches and Debates of Senator Joe McCarthy Delivered in the United States Senate, 1950–1951 |publisher = Gordon Press |year= 1951 |pages = 264, 307, 215 |isbn = 0-87968-308-2}}. See also [[w:Joseph McCarthy#McCarthy and the Truman administration]].</ref> Near the end of 1953, McCarthy began referring to "twenty-''one'' years of treason" to include Eisenhower's first year in office.<ref>{{cite book|last = Fried |first = Albert |title = McCarthyism, The Great American Red Scare: A Documentary History |publisher = Oxford University Press |year = 1996 |page = [https://archive.org/details/mccarthyismgreat00frie/page/179 179] |isbn = 0-19-509701-7 |url = https://archive.org/details/mccarthyismgreat00frie/page/179 }}. See also [[w:Joseph McCarthy#McCarthy and Eisenhower]].</ref> In that political environment, Eisenhower doubtless knew in early 1954 that it would be very difficult for him politically if the Communist Ho Chi Minh actually won elections in Vietnam scheduled for July 1956, while Eisenhower would likely be running for reelection. He therefore worked clandestinely behind the scenes to help replace the unpopular Vietnamese Emperor {{w|Bảo Đại}} with {{w|Ngo Dinh Diem}}<ref name=Moyar>Moyar, Mark (2006). Triumph Forsaken: The Vietnam War, 1954–1965. New York: Cambridge University Press, pp. 41–42. See also [[w:Ngo Dinh Diem#Becoming Prime Minister and consolidation of power]].</ref> following the [[w:Geneva Conference (1954)|Geneva Accords of 1954]]. Diem effectively canceled the reunification elections scheduled for 1956 as part of those agreements.<ref name=Moyar/> A decade later, during the 1964 US presidential election campaign, President {{w|Lyndon Johnson}} was similarly being accused of being "soft on Communism" by his Republican opponent, {{w|Barry Goldwater}}. Johnson worked clandestinely to provoke North Vietnam into attacking US military vessels in the [[w:Gulf of Tonkin|Gulf of Tonkin]]. When the US Navy destroyer [[w:USS Maddox|USS ''Maddox'']] fired at "false radar images" on August 4, 1964, Johnson got the US Senate to approve the {{w|Gulf of Tonkin Resolution}} effectively giving Johnson a blank check to escalate the US war in Vietnam to counter this "unprovoked" attack. Only two US Senators voted against that resolution, and both were defeated in the next election. Johnson ultimately complained that the war had killed “the woman I really loved — the Great Society”.<ref><!-- How Vietnam Killed the Great Society -->{{cite Q|Q108895834}}</ref> [[w:United States involvement in regime change|The media environment drove US policy towards Vietnam from the end of world War II]] to the {{w|Fall of Saigon}} in 1975. The factual justification for the War on Terror, as documented in the present article, seems much less substantial than the justification for the Vietnam War. On May 23, 2013, then-US President Obama noted that terrorism caused fewer American deaths than car accidents or falls in the bathtub. He occasionally ''had to be badgered by advisors into choices commensurate with popular fear.'' He worried, too, that counterterrorist priorities “swamped” his other foreign policy aspirations.<ref><!-- Humane: How the United States Abandoned Peace and Reinvented War -->{{cite Q|Q108896140}}, pp. 268, 299-300.</ref> *''US military operations during the Vietnam War and the War on Terror seem primarily to have been driven by information the mainstream US media chose to highlight, selected to please their major funders, while routinely suppressing information that conflicted with that image. Many people were killed, because the different parties to conflict had (a) conflicting perceptions of reality and (b) inadequate understanding of the power of nonviolence and community policing.''<ref>See also the Wikiversity article on "[[confirmation bias and conflict]]".</ref> An October 14, 2014, story quoted then-President Obama as saying, “Very early in [the discussions about helping Syrian rebels], I actually asked the C.I.A. to analyze examples of America financing and supplying arms to an insurgency in a country that actually worked out well. And they couldn’t come up with much.”<ref>{{Citation | last = Mazzetti | first = Mark | date = October 14, 2014 | title = C.I.A. Study of Covert Aid Fueled Skepticism About Helping Syrian Rebels | newspaper = New York Times | url = https://www.nytimes.com/2014/10/15/us/politics/cia-study-says-arming-rebels-seldom-works.html?_r=0 | accessdate = 2017-03-09}}</ref> * ''If the overall record of US foreign interventions is positive, the successes are well hidden.'' === 2.12. G. W. Bush: "Why do they hate us?" === On 2001-09-20, nine days after the {{w|September 11 attacks}}, US President {{w|George W. Bush}} asked, "Why do they hate us?"<ref name='Why'>{{cite Q |Q61743801 }}</ref> He gave his own answer to this rhetorical question: "They hate [our] democratically elected government. Their leaders are self-appointed. They hate our freedoms: our freedom of religion, our freedom of speech, our freedom to vote and assemble and disagree with each other." We need serious, unclassified research into why people choose one side or the other in this and other conflicts, why some people remain on the sidelines, and why some change their affiliations over time, increasing or decreasing their level of support, deserting or defecting. Such research might identify a large portion of US enemies motivated a desire for the freedoms the US claims to hold so dear and a hatred of US support for governments that deny them those freedoms.<ref>See comments about Thomas Carothers in {{cite Q |Q61754939 }}</ref> And in addition to researching such questions, we need a media system that will disseminate the results, free from the conflicts of interest that encumber the mainstream media virtually everywhere today, as suggested elsewhere in this essay. In fact, US governmental officials knew on September 11, 2001, that the government of Saudi Arabia was involved in the preparations for the September 11 attacks. This is documented in [[w:The 28 Pages#Declassification|"The 28 Pages"]] of material redacted from the December 2002 report of the joint US House and Senate Committee investigating intelligence failures regarding the September 11 attacks, removed on the insistence of US President George W. Bush. Those documents do not say whether President Bush himself knew that when the [[w:United States invasion of Afghanistan|US, the UK, Canada and Australia invaded Afghanistan 2001-10-07]]. However, before the US invaded, the government of Afghanistan offered to turn over Osama bin Laden, but they wanted evidence of bin Laden's complicity in the {{w|September 11 attacks}}. Evidently, the US invaded Afghanistan for other reasons. And the failure of the mainstream media in the US to support the Afghani request for evidence of bin Laden's complicity and their virtually nonexistent coverage of "{{w|The 28 Pages}}" seems to support the assertions elsewhere in this essay. * The rules of evidence in the court of public opinion are whatever will maximize the power of those who control media funding and governance. The future prospects for peace on earth would be enhanced greatly by (a) research into what motivates people in conflict and (b) media that would be more likely to tell the public what they need to know to understand their opposition in conflict. === 2.13. Obama's approach to counterterrorism === On December 6, 2016, Obama gave his last foreign policy address as President. He had seven major points: # Keep the threat in perspective. # Don't overreach. # Respect rule of law. # Fight terrorists in a way that does not create more. # Insist on transparency and accountability not just in times of peace but, more importantly, in times of conflict. # Emphasize diplomacy. # Uphold the civil liberties that define us. These points seem to summarize the thrust of this essay up to this point. The recent spike in terrorist deaths in Figures 1 and 2 and in Appendix 1 all seem to be products of violations of these principles.<ref>{{Citation | last = Obama | first = Barack | date = 2016-12-06 | title = Remarks by the President on the Administration's Approach to Counterterrorism | publisher = Obama White House | url = https://obamawhitehouse.archives.gov/the-press-office/2016/12/06/remarks-president-administrations-approach-counterterrorism | accessdate = 2017-03-14}}</ref> === 2.14. Eli Lake on "How Trump could finally win the war on terror" === Three days after Obama's December 6 speech, Eli Lake acknowledge that what Obama said makes good sense for the most part. Lake continued by saying that the failure of G. W. Bush and Obama to win the war on terror is due to a failure to acknowledge that jihadists "seek conquest." They don't hate us because of our freedom, as President George W. Bush claimed. "Their objective is not to provoke an overreaction where America ceases to be a democracy. ... These groups want to force the non-Muslim world ... to submit to Islamic rule."<ref>{{Citation | last = Lake | first = Eli | date = 2016-12-09 | title = Commentary: How Trump could finally win the war on terror | newspaper = Chicago Tribune | url = http://www.chicagotribune.com/news/opinion/commentary/ct-trump-terrorism-islamic-state-20161209-story.html | accessdate = 2017-03-14}}</ref> The analysis here does not contradict Lake's claim regarding the goal of some of the Jihadists' leadership. However, it's not clear why that's even relevant. If you, dear reader, know of any evidence why it should make a difference, please post it here. We next consider how the structure of the media contribute to the escalation and perpetuation of conflict, and how alternative systems for funding and governing media might improve the prospects for conflict resolution and world peace. == 3. Media == This section first discusses research on human psychology and how that interacts with the political economy of the media. This suggests that winning the War on Terror might require greater democratic control of the media. Alternative systems for media funding and governance are then reviewed. === 3.1. Human psychology and the media === This section discusses two important observations by [[w:Daniel Kahneman|Daniel Kahneman]], mentioned above: # Humans tend to be excessively overconfident in the value of what they think they know. # People pay too much attention to things that are novel and poignant and too little attention to things that are more important and more common. ==== 3.1.1. Overconfidence ==== Most humans tend to remember news and information that is consistent with their preconceptions and forget or don’t even see conflicting evidence.<ref name=Kahneman2011>Kahneman (2011)</ref> To counteract this Kahneman pushes us to be more humble about what we think we know: We should look for credible information sources that conflict with our preconceptions.<ref>One of Kahneman's (2011) major themes is that nearly everyone tends to overestimate the value of current knowledge and underestimate how far wrong their preconceptions likely are. This approach works fine for most situations -- and helps us avoid wasting time searching for better answers to unimportant questions. However, it tends to produce poor assessments of some of the most important situations we encounter. We could often arrive at much better decisions if we pushed ourselves to identify really important issues and look harder for contrary information for those cases.</ref> If we do, we may find with the famous [[w:Pogo (comic strip)|comic strip character Pogo]] that, “We have met the enemy, and he is us.” * ''How easy it is to make people believe a lie, and [how] hard it is to undo that work again! ''<ref>{{Citation | last = Twain | first = Mark | editor-last = Griffin | editor-first = Benjamin | editor2-last = Smith | editor2-first = Harriet Elinor | year = 2013 | title = Autobiography of Mark Twain, Vol. 2 | page = 302 | url = https://en.wikiquote.org/wiki/Mark_Twain | accessdate = 2017-02-17}}</ref> * ''I’m hurt less by things I don’t know than things I do know that ain’t so.''<ref>Richard Salmon, personal communication, circa 1973.</ref> ==== 3.1.2. Novel and poignant ==== This essay began with a discussion of the minuscule nature of terrorism compared to other causes of death. This raises the following question: * ''Why do we place so much more emphasis on terrorism than on other issues that are much more common causes of death?'' [[w:Daniel Kahneman|Kahneman]] says that people pay more attention to things that are novel and poignant, like terrorism incidents. As a result, media organizations look for that kind of material. This becomes a problem when reports lack adequate context, thereby leading the public to believe that problems highlighted are far worse than they really are. For example, "[[stroke]]s cause almost twice as many deaths as all accidents combined, but 80% of respondents [in a survey] judged accidental death to be more likely. ... [This is because media] coverage is itself biased toward novelty and poignancy. The media do not just shape what the public is interested in, but also are shaped by it."<ref name=Kahneman2011/> ==== 3.1.3. Availability cascade vs. media feeding frenzy ==== This discussion of novelty and poignancy helps explain the phenomenon of a self-reinforcing cycle of high public interest in a certain topic that invites the media to produce a series of stories about that issue. The resulting cycle is sometimes called an [[w:Availability cascade|availability cascade]].<ref>Kahneman (2011)</ref> However, an [[w:Availability cascade|availability cascade]] rarely occurs when it might displease someone with substantive control over the media, discussed in the next section on the “political economy of the media.” The term "[[w:Media feeding frenzy|media feeding frenzy]]" is almost synonymous with “availability cascade,” but "media feeding frenzy" may suggest more of a role for flinching, shading or even suppressing a story or limiting its run to minimize displeasure to media owners, managers, or funders.<ref>See the discussion of the 5-fold increase in the incarceration rate in the US over the past 40 years accompanying Figure 9 in this essay.</ref> === 3.2. The political economy of the media === * ''Media organizations sell changes in the behaviors of their audience to their funders.'' A media organization without an audience won’t have funding for long. If the audience fails to change behaviors in ways that please the funders -- or, worse, if they change behaviors in ways that threaten the funders -- the money will go elsewhere. A media organization must please both its audience and its funders. With commercial media, story coverage is sometimes "tailored to maximize its appeal to key demographic groups: those who are most likely to buy the advertised product. When target audiences place low value on hard news, media outlets have an incentive to reduce current affairs and political reporting in favor of entertainment and sports coverage."<ref>{{Citation | last = Ognyanova | first = Katherine | editor-last = Lloyd | editor-first = Mark | editor2-last = Friedland | editor2-first = Lewis A. | year = 2016 | title = The Communications Crisis in America, And How to Fix It | chapter = Researching Community Information Needs | publisher = Palgrave Macmillan | isbn = 1-349-95030-0}}</ref> Tailoring news to sell products is serious but minor relative to suppressing coverage of favors that major advertisers get from government, which is the primary activity of legislators, at least in the US Congress, according to [[w:Lawrence Lessig|Lawrence Lessig]]'s [[w:Republic, Lost|''Republic, Lost'']]. It is also minor when compared to suppressing information regarding likely US actions against democracy in foreign countries or stampeding the US into war on questionable grounds, discussed elsewhere in this essay. Moreover, the [[w:National Association of Broadcasters|National Association of Broadcasters]] vigorously opposed FCC support for research into the critical information needs of different communities in the US.<ref>Lloyd and Friedland (2017, p. 14)</ref> Why would they do that? Are they afraid that the research might suggest that the public has information needs that are not being met and might lead to efforts to fix that problem? ==== 3.2.1. Media coverage of conflict ==== As conflicts escalate, at some point media organizations become party to the conflict, amplifying the propaganda that drives people apart and reduces the chances for negotiated settlements and resolution using law enforcement. This is more subtle and more destructive than the standard dictum that, "Truth is the first casualty of war."<ref>The first use of that phrase in this form appears to have been by [[w:Philip Snowden, 1st Viscount Snowden|Philip Snowden, 1st Viscount Snowden]].</ref><ref>{{Citation | last = Knightley | first = Phillip | year = 2004 | origyear = 1975 | title = 'The First Casualty: The War Correspondent as Hero and Myth-Maker from the Crimea to Iraq | edition = 3rd | publisher = Johns Hopkins U. Pr. | isbn = 0801880300}}</ref> We consider three examples: The [[w:American Civil War|U.S. Civil War]], the [[w:Cold War|Cold War]], [[w:Israel|Israel]] today, and the US since the September 11 attacks. By the time of the [[w:American Civil War|U.S. Civil War]], many moderately sized cities in the US had at least two newspapers, often with very different political perspectives. As the South began to succeed, some papers in the North recommended that the South should be allowed to leave. “The government, however, was not willing to allow 'sedition' to masquerade (in its opinion) as 'freedom of the press.'” Several newspapers were closed by government action. After the massive Union defeat at the [[w:First Battle of Bull Run|First Battle of Bull Run]], angry mobs in the North destroyed substantial property used by “successionist” newspapers. Those still in publication quickly came to support the war to avoid mob action and government repression and to retain their audiences.<ref>Harris (1999, esp. ch. 8, pp. 97-107)</ref> Given what is known today about the evolution of conflict (including the research on the long-term impact of alternative approaches to conflict discussed above), it seems likely that nearly everyone would be better off today if the North had let the South succeed. This is consistent with the findings of Chenoweth and Stephan, discussed above. It is also supported by the fact that the South had a substantial population of free whites, many of whom did not like having to compete with slave labor and might have supported slaves fleeing to the Union without Union troops on Confederate soil.<ref>{{Citation | last = Gillespie | first = Michele | year = 2004 | title = Free labor in an unfree world: White artisans in slaveholding Georgia, 1789-1860 | publisher = U. of Georgia Pr. | isbn = 0820326704}}</ref> The violence of the Civil War built (or at least strengthened) bonds between poor Southern whites and the Southern aristocracy that contribute to the problems with racism that plague the US to this day.<ref>This was written 2017-03-05. The violence of the [[w:American Revolution|American Revolution]] and the role of race in that conflict also contributed to the current racism in the US in the same way, though to a lesser extent.</ref> Many of the questionable actions of the [[w:Cold War|Cold War]] can be explained as heavily influenced by mainstream media support for US international business interest. For example, if major oil companies in the US did not advertise, might more questions have been raised about the destruction of democracy in Syria in 1949 and Iran in 1953, at the behest of international petroleum interests? If United Fruit did not advertise, might more questions have been raised about the 1954 coup in Guatemala? The tie is less specific regarding the cancellation of elections in Cuba in 1952 or Vietnam in 1956, but if the mainstream commercial media in the US had raised too many questions about those events, they likely would have offended executives in many multinational corporations. If all of Eisenhower's contacts "knowledgeable in Indochinese affairs" agreed that Ho Chi Minh would likely have gotten 80 percent of the popular vote in 1954, the mainstream commercial media in the US should have been aware of that; the fact that substantive questions were not raised about the cancellation of elections there in 1956 strongly suggests editorial decisions to suppress that coverage. Similar claims can be made about the destruction of democracy in Brazil in 1954 and Chile in 1973. This may not qualify as proof beyond a reasonable doubt, [[w:United States constitutional criminal procedure|required for a criminal conviction]]. However, it would seem to meet the standard of a [[w:Burden of proof (law)#Preponderance of the evidence|"preponderance of the evidence']], required in a civil trial. The evolution of the media in Israel followed roughly the development of the US media, according to Israeli scholar Yorim Peri:<ref>Peri (2012)</ref> At the creation of the state of Israel, newspapers tended to be associated with political parties. In the late 1960s and early 1970s, they became more commercial and professional.<ref>Peri (2012) noted that many of the journalists joining the profession starting especially in the late 1960s had been trained in the US and followed US journalistic practices to a large extent.</ref> However, the range of acceptable political discourse has always been constrained by concerns about national security. “When the security situation is tense, pressure for consensus and uniformity tends to increase. At such times, the audience is less willing to hear different opinions. Therefore the media cannot completely fulfill its function as the arena where issues are hashed out or hammered out before being brought to the political system for a policy decision. An ongoing state of emergency undermines the readiness for pluralism, tolerance and liberalism and amplifies public expectations that the media will exhibit more ‘social responsibility’ -- be less critical, more committed to the collective endeavor, and more supportive of the national leadership. Above all, a state of emergency legitimizes the state’s deeper and deeper intrusion into the private sphere and into civil society.” Peri further claimed that the Israeli media could have warned of the impending attack prior to the initiation of the 1973 [[w:Yom Kippur War|Yom Kippur War]]. After the war “Israeli journalists conceded that their total dependence and trust in the government and uncritical adoration of the top brass were responsible for media not issuing a warning that war was about to break out.”<ref>Peri (2012, pp. 22-23)</ref> Peri continued, “In the 1990s -- during the peace process, which made it appear that the era of warfare was at an end and that Israel was becoming a postwar society -- the professional autonomy of the media grew, and journalists adopted a more critical stance. However, the failure of the peace talks in the summer of 2000 and the outbreak of the second Intifada with its suicide attacks aimed at the heart of the civilian population led to a serious retreat ... . State agencies and the public even more so again exerted pressure for media reorientation, demanding that the media restrain its criticism and circle the wagons."<ref>Peri (2012, p. 23)</ref> Peri’s claims are consistent with an earlier paper on “Palestinian civil resistance against Israeli military occupation,” which claimed that the ''nonviolence'' of the [[w:First Intifada|First Intifada]] made a greater contribution to the ability of Palestinians to live and prosper in that region than anything Palestinians have done before or since.<ref>{{Citation | last = King | first = Mary Elizabeth | editor-last = Stephan | editor-first = Maria J. | year = 2009 |title = Civilian Jihad: Nonviolent struggle, democratization, and governance in the Middle East | publisher = Palgrave MacMillan | pages = 131-155 | chapter = chapter 10. Palestinian civil resistance against Israeli military occupation | isbn = 978-0-230-62141-1}}</ref><ref>See also Chenoweth and Stephan (2011, pp. 119-120, 138, 145)</ref> [[File:Shootings as a percent of all incidents during the First Intifada.svg|thumb|Figure 8. Shootings as a percent of all incidents during the First Intifada.<ref>Chenoweth and Stephan (2011, p. 120)</ref>]] The First Intifada began spontaneously after four Palestinians were killed and eight seriously injured after an Israeli military vehicle struck a car carrying Palestinian day laborers on December 7, 1987.<ref>Chenoweth and Stephan (2011, p. 123)</ref> This led to 65,661 nonviolent protests and 140 shooting incidents in 1988 and 1989. Overreaction by Israeli troops and settlers over the first 18 months led to the deaths of roughly 650 Palestinians, totally out of proportion to the physical threat.<ref>Chenoweth and Stephan (2011, p. 120, Table 5.1)</ref> Press coverage led to condemnation of this overreaction in Israel and around the world. [[w:Yitzhak Rabin|Yitzhak Rabin]] was elected Prime Minister of Israel in 1992, promising negotiations with Palestinians, defeating the more hawkish [[w:Yitzhak Shamir|Yitzhak Shamir]]. While nearly everyone in occupied Palestine understood they could not win with guns, those advocating nonviolence were unable to prevent everyone under occupation from using firearms, as quantified in Figure 8: Shootings a a proportion of total incidents were only one sixth of a percent in 1988 and rose to four thirds of a percent in 1992; they averaged half a percent from 1988 to 1992. The nonviolent advocates were also unable to keep youth from throwing rocks, which many supporters of Israel did not see as nonviolent. Moreover, Israel was clandestinely arming [[w:Hamas|Hamas]], as a counterweight to both the nonviolence and the [[w:Palestinian Liberation Organization|Palestinian Liberation Organization]] (PLO). And many PLO leaders were still advocating violence. Chenoweth and Stephan classified the First Intifada as a "partial success," because it obtained some concessions including international recognition but failed to shake off the occupation.<ref>Chenoweth and Stephan (2011, p. 145)</ref> Unfortunately, too few Palestinians recognized what they had gained through nonviolence. On September 28, 2000, [[w:Ariel Sharon|Ariel Sharon]] visited the Temple Mount complex, the holiest place in the world to Jews and the third holiest site in Islam, accompanied by an escort of over 1,000 Israeli police officers. This was seen as a deliberate provocation by many and produced a violent response by Palestinians. That violence seemed to help Sharon win the election for Prime Minister the following February 6 with 62 percent of the vote. The research on nonviolence summarized above and elsewhere suggests that if the Palestinian response had been nonviolent, it could have helped bridge rather than deepen the gap between Jews and Palestinians. To what extent might Peri’s comments apply to the US response to the [[w:September 11 attacks|September 11, 2001]]? This includes the suppression of debate in the mainstream commercial media over the official justification for the 2003 US-led invasion of Iraq. It also includes the creation of the [[w:Homeland Security|US Department of Homeland Security]] and the expansion of surveillance activities by agencies like the [[w:National Security Agency|US National Security Agency (NSA)]] without authorization from Congress. This was exposed by [[w:Edward Snowden|Ed Snowden]]<ref> {{Citation | title = Edward Snowden | publisher = Wikipedia | url = https://en.wikipedia.org/wiki/Edward_Snowden | accessdate = 2017-02-17}}</ref> following the apparent perjury of [[w:James R. Clapper|James Clapper]], [[w:Director of National Intelligence|Director of National Intelligence]], before a [[w:United States Senate Select Committee on Intelligence|US Senate Select Committee on Intelligence]].<ref> {{Citation | title = James Clapper | publisher = Wikipedia | url = https://en.wikipedia.org/wiki/James_Clapper | accessdate = 2017-02-17}}</ref> In the three examples of this section, the US Civil War, Israel and the US since September 11, 2001, the media arguably became a party to the conflict. In the first two cases, their audiences seemed to demand it. In the latter two cases, there’s substantial evidence that the general interests of the bottom 99 percent of the people in Israel and the US were ill served by the following: # Intelligence services that either failed to appropriately assess potential threats or suppressed information or fabricated evidence to please superiors. # Excessive willingness of the media to support unquestioningly the policies and pronouncements of those leading the national security apparatus.<ref>It is now known that members of the Saudi royal family and employees of the Saudi embassy and consulates in the US helped the suicide mass murderers of September 11, 2001, get training in the US to help them do what they did on that fateful day. This is documented in [[w:The 28 Pages|"The 28 Pages,"]] which the George W. Bush administration insisted were classified and were therefore not published with the rest of the December 2002 report of the [[w:Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001|Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001]] and were largely declassified in July 2016. This documents that the Bush administration knew before it invaded Iraq, and possibly before it invaded Afghanistan, that 9-11 was supported by high-level Saudis, in addition to the information available shortly after 9-11 that 15 of the 19 suicide mass murderers of Sept. 11 were Saudis. So why did the US NOT invade Saudi Arabia and instead invaded Afghanistan and Iraq? And why did the mainstream US media so eagerly support the invasions of Afghanistan and Iraq in 2001 and 2003? And why has the mainstream US media NOT made an issue of the new information in "The 28 Pages" released in July 2016? The thrust of the present article is that several factors contribute to this documented record of media failure, one of which is doubtless the fact the big oil companies advertise and have had great relations with the Saudi royal family dating back to the 1930 -- and the Afghanis had refused to approve the [[w:Turkmenistan–Afghanistan–Pakistan–India Pipeline|construction of a pipeline on their soil, finally begun on December 13th, 2015]].</ref> For more, see the discussion in “Implications,” below. Let’s now turn from audience to funding and management of media organizations. ==== 3.2.2. Media ownership, funding and profitability ==== * ''Media organizations can libel and slander poor people with impunity but must of necessity flinch before disseminating anything that might offend anyone with substantive control over the media.'' The basic economics of journalism has multiple consequences for efficiency: * Stories impacting poor people can be disseminated with little or no fact checking, because their ability to retaliate is minimal. * Stories that might offend anyone with substantive influence over the media require much more fact checking and editing that might otherwise be required and may never appear.<ref name=Focke/> [[File:U.S. incarceration rates 1925 onwards.png|thumb|350px|Figure 9. A graph of the incarceration rate under state and federal jurisdiction per 100,000 population 1925–2008 (omits local jail inmates). The '''male incarceration rate''' (''top line'') is roughly 15 times the '''female rate''' (''bottom line'').]] Evidence of this is seen in the five-fold increase in the [[w:United States incarceration rate|incarceration rate in the US]] between 1975 and 2000, after the incarceration rate had been relatively stable at roughly 0.1 percent for the previous half century; see Figure 9.<ref name=incar>{{Citation | title = United States incarceration rate | publisher = wikipedia | url = https://en.wikipedia.org/wiki/United_States_incarceration_rate | accessdate = 2017-02-26}}</ref> The obvious driver of this was a shift in US politics that began around 1975 to "get tough on crime." This political change was driven by a shift in editorial policies of mainstream commercial broadcasting to focus on the police blotter.<ref>e.g., Sacco (2005) and others cited in [[w:United States incarceration rate|incarceration rate in the US]].</ref> The broadcasters found they could reduce expenditures for investigative journalism, thereby reducing the risks of offending major advertisers,<ref name=Focke/> while still retaining (and perhaps increasing) their audience. * [[w:Fascination with death|''If it bleeds, it leads.'']]<ref>This is consistent with the discussion of "availability cascade," as discussed by Kahneman (2011, pp. 142-145) and summarized above.</ref> The public got the impression that crime was out of control, even though no increase in crime was evident in the best available data. Politicians who wanted to "get tough on crime" replaced those who resisted this trend. The laws were changed, and the [[w:United States incarceration rate|incarceration rate in the US]] jumped dramatically, especially among people of color.<ref>{{cite book |last=Alexander |first=Michelle |year=2010 |title=The New Jim Crow: Mass Incarceration in the Age of Colorblindness |publisher=The New Press |location=New York |isbn=978-1-59558-103-7 }}</ref> Beyond this, what happens when police and prosecutors get convictions based on torture, coerced perjury, planted or falsified evidence or suppression of exculpatory evidence? Because stories from the police blotter are so cheap to produce, journalists and media outlets have a conflict of interest in honest reporting on any case involving official misconduct unless it becomes so big the media would lose audience for failing to report it.<ref name=incar/> This provides an opportunity for corrupt police, prosecutors and judges, who believe they get promoted on convictions. They get credit for fighting crime without actually impacting the crime rate, because the real perpetrators are still free. When fraudulent convictions are obtained disproportionately against [[w:Tom R. Tyler|minorities, increasing portions of minority communities come to distrust the police]]. This in turn makes it more difficult to obtain the cooperation of the community, thereby making policing more difficult.<ref>Tyler and Huo (2002)</ref> The law enforcement budget is safe, primarily because the media continue to report crimes committed by poor people with few, if any, references to the fact that there has been no substantive change in crime to [[w:United States incarceration rate|support the changes in law that have driven up the incarceration rate documented in Figure 9]]. Between 1989 and March 2017 at least 15 major [[w:police misconduct|police scandals]] came to light involving over 1,800 innocent defendants convicted on planted or falsified evidence, forced confessions, coerced perjury, suppression of exculpatory evidence, inadequate defense, and other forms of official misconduct by police, prosecutors and judges -- who evidently believe that they get promoted based on convictions.<ref>The 2016 annual report of the [[w:National Registry of Exonerations|National Registry of Exonerations]] reported, "1,994 known exonerations in the United States since 1989 (as of February 26, 2017)." These 1,994 cases do not include over 1,800 defendants cleared in 15 large-scale [[w:police misconduct|police scandals]]. {{Citation | date = March 7, 2017 | title = Exonerations in 2016 | publisher = National Registry of Exonerations, Newkirk Center for Science and Society, U. CA, Irvine | url = http://www.law.umich.edu/special/exoneration/Documents/Exonerations_in_2016.pdf | accessdate = 2017-03-17}} and {{Citation | last = Gross | first = Samuel R. | last2 = Possley | first2 = Maurice | last3 = Stephens | first3 = Klara | date = March 7, 2017 | title = Race and wrongful convictions in the United States | work = report | publisher = National Registry of Exonerations, Newkirk Center for Science and Society, U. of CA Irvine | url = http://www.law.umich.edu/special/exoneration/Documents/Race_and_Wrongful_Convictions.pdf | accessdate = 2017-03-17}}</ref> It would be interesting to study how this might be different with citizen-directed subsidies for media, as discussed below. If the research summarized in this section is accurate and balanced, then citizen-directed subsidies for media might provide * More coverage of crimes committed by major advertisers and earlier exposure of official misconduct by police, prosecutors and judges, and * Fewer people sentenced to death or long terms in prison for crimes they did not commit.<ref name=incar/> Other examples of how news can be distorted by powerful people were described by [[w:Charles Lewis (journalist)|Charles Lewis]] in his book ''935 Lies.''<ref>Lewis (2014)</ref> He left the [[w:CBS News|CBS news]] program [[w:60 Minutes|''60 Minutes'']] in 1989 primarily over two concerns: * Frustration with the meddling of a CBS executive in stories of great importance to the nation, and * The fact that CBS was firing senior investigative journalists at that time.<ref>Lewis (2014, esp. pp. 132-135, 196-198)</ref> One of his examples was a discussion of “Tobacco on Trial”.<ref>{{Citation | date = January 3, 1988 | title = Tobacco on Trial | series = 60 Minutes | publisher = CBS News | url = http://www.cbsnews.com/videos/tobacco-on-trial/ | accessdate = 2017-02-22}}</ref> This story was of particular concern to [[w:Lawrence Tisch|Lawrence Tisch]], President and Chief Executive Officer (CEO) of CBS at the time. Tisch was also a co-founder and major stockholder in Loews, which owned Lorillard Tobacco, a party to a lawsuit discussed in “Tobacco on Trial.” *''Journalism is spreading what someone does not want you to know; the rest is propaganda.''<ref>[[w:Horacio Verbitsky|Horacio Verbitsky]]</ref> This episode of ''60 Minutes'' was one small battle pitting honest journalism against the survival and profitability of the US tobacco industry. A few years later, in January 1994, a former R. J. Reynolds employee with a doctorate in engineering began talking secretly to the Food and Drug Administration and to journalists. She exposed how the tobacco industry manipulated nicotine levels to maximize addiction and company profitability while also “quietly nudging US Department of Commerce and US trade representatives to spend millions of taxpayer dollars to pry open foreign markets to American cigarettes”. After some of this aired on [[w:ABC News|ABC’s]] [[w:Day One (TV news series)|''Day One'']], [[w:Philip Morris|Philip Morris]] filed a $10 billion libel lawsuit against ABC and the show’s lead producers for what they claimed was “false and defamatory” reporting that had been produced “knowingly, recklessly, and with malice.” ABC had already invested half a million dollars in an expanded investigation of this, which was ultimately canceled to end the litigation. “Philip Morris had shown that ‘for a paltry $10 million or $20 million in legal fees … you can effectively silence the criticism,’” Lewis wrote.<ref>Lewis (2014, esp. pp. 136-139)</ref> The decision to cancel that show was a sensible business decision on the part of ABC: They had much less at stake than Philip Morris, whose entire future was threatened. It was highly unlikely that ABC could ever gain an increase in audience sufficient to cover their legal costs in this [[w:Strategic lawsuit against public participation|Strategic lawsuit against public participation (SLAPP)]]. Meanwhile, [[w:Philip Morris International|Philip Morris]] also got what they wanted in international trade agreements. They’ve used these trade agreements to sue the governments of Uruguay, Australia, and Norway for lost profits due to labeling requirements in those countries that have led to reductions in tobacco consumption (and improvements in public health). The more stringent tobacco labeling requirements in Uruguay have improved public health there, but the legal fees are a major drain on the government's budget. Fortunately, former New York Mayor Bloomberg and the Bill and Melinda Gates Foundation launched a multi-million dollar fund to help smaller countries (including Uruguay) fight legal battles with tobacco companies. Bloomberg said, "We are in this to help countries that can't afford to defend themselves against an industry which will try to kill a billion people this century,"<ref>{{Citation | last = Davies | first = Wyre | date = 7 April 2015 | title = Michael Bloomberg fights big tobacco in Uruguay | publisher = BBC News | url = http://www.bbc.com/news/world-latin-america-32199250 | accessdate = 2017-03-17}}</ref> * ''With more citizen direction in the selection of news, would the laws have been written to allow Philip Morris to file suit in cases like this?'' What does this discussion of incarcerations and tobacco say about how the Middle East might be different without the impact of big oil or other major international businesses with major advertising budgets? Would the United States have supported the Saudi royal family since the 1930s if major US advertisers did not believe they benefitted from maintaining the power of the House of Saud?<ref>[[w:House of Bush, House of Saud|Without a business relationship dating back over 30 years between the Bush family and the House of Saud]], would the George W. Bush administration have turned a blind eye to Saudi complicity in the September 11 attacks and diverted attention instead to Afghanistan and Iraq, even though neither seemed to have been complicit in the event?</ref> ==== 3.2.3. Media and politics ==== The mainstream media interacts with politics as follows: * ''The mainstream media create the stage upon which politicians read their lines.''<ref>For example, [[w:David Frum|David Frum]], former speechwriter for [[w:George W. Bush|George W. Bush]], has also said, "Republicans originally thought that Fox worked for us and now we're discovering we work for Fox." {{Cite news |publisher=ABC |url=http://blogs.abcnews.com/nightlinedailyline/2010/03/david-frum-on-gop-now-we-work-for-fox.html |date=March 23, 2010 |title=David Frum on GOP: Now We Work for Fox |work=Nightline}} See also the Wikipedia article on "[[w:Fox News controversies]]".</ref> By selective coverage, the mainstream media can paint the black white and the white black. * ''The mainstream media [[w:Overton window|define the range of acceptable political discourse.]]'' [[w:Campaign finance in the United States#Impact of finance on the results|In 93 percent of the 1,349 races in the US House and Senate in 2012, 2014, and 2016, the candidate with the most financial support won]].<ref>This seems more like [[w:Plutocracy|plutocracy]] than democracy.</ref><ref name="crp2016">{{cite web | last1=Balcerzak | first1=Ashley | title=Where the money came from, not how much, mattered in the presidential race | url=https://www.opensecrets.org/news/2016/11/where-the-money-came-from-not-how-much-mattered-in-the-presidential-race/Balcerzak | website=OpenSecrets.org | publisher=Center for Responsive Politics | accessdate=2017-02-21 | date=November 9, 2016 }}</ref> This disturbing measure of the outsized impact of funding on elections is a product of the interaction between the business model of the media and the fact that we humans too often make decisions based on what comes readily to mind -- and too seldom check our facts, as outlined in the discussion above of Kahneman's research. This is supported by the virtual elimination of investigative journalism from US television by the early 2000s, except for a few popular shows like [[w:60 Minutes|''60 Minutes.'']]<ref><!-- McChesney (2004) The Problem of the Media: U.S. Communication Politics in the 21st Century -->{{cite Q|Q7758439|page=81}}.</ref> Investigative journalism is expensive and risky, as noted in the previous section on “Media ownership, funding and profitability”. Fact checking during political campaigns could make it easier for politicians to get elected with less advertising. That’s a losing business from at least two perspectives: # Politicians and their big money supporters, whose questionable claims were exposed, would be offended by being challenged. # Voters could make more intelligent decisions in the voting booth with less advertising purchased by candidates. Most journalist working for the mainstream broadcasters in the US today have little time to check facts, and are largely encouraged to let people state their positions without question. An exception is [[w:Democracy Now!|''Democracy Now!'']], where hosts like [[w:Amy Goodman|Amy Goodman]] frequently ask guests for their response to what their opposition says. This lack of fact checking has contributed to the current [[w:post-truth politics|post-truth era]] in the US, where political discourse is increasingly driven by [[w:fake news|fake news]] circulating in part on social media. By some accounts, the success of the [[w:Brexit|Brexit referendum]] in the UK and the Trump candidacy in the 2016 US presidential election were build in part on highly successful placement of claims in social media selected to be credible to a specific audience based on data mining of people’s online activities. A leader in this field was for a time [[w:Cambridge Analytica|Cambridge Analytica]], whose Chief Executive, Alexander Nix, said: * "Today in the United States we have somewhere close to four or five thousand data points on every individual. ... So we model the personality of every adult across the United States, some 230 million people."<ref name="20161021sky">{{citation |url=http://news.sky.com/story/behind-the-scenes-at-donald-trumps-uk-digital-war-room-10626155 |title=Behind the scenes at Donald Trump's UK digital war room |last1=Cheshire |first1=Tom |date=21 October 2016 |publisher = [[w:Sky News|Sky News]] |archive-url=https://web.archive.org/web/20161021180327/http://news.sky.com/story/behind-the-scenes-at-donald-trumps-uk-digital-war-room-10626155 |archive-date=21 October 2016}}</ref> On May 1, 2018, Cambridge Analytica officially ceased operations. In doing so, it maintained its innocence, claiming it "has been vilified for activities that are not only legal, but also widely accepted as a standard component of online advertising in both the political and commercial arenas."<ref>{{Cite news |url=https://www.engadget.com/2018/05/02/cambridge-analytica-is-shutting-down-following-facebook-scandal/ |title=Cambridge Analytica is shutting down following Facebook scandal |last=Lumb |first=David |date=2 May 2018 |work=[[Engadget]] |access-date=2 May 2018 }}</ref><ref>{{cite web |title=Cambridge Analytica and Scl Elections Commence Insolvency Proceedings and Release Results of Independent Investigation into Recent Allegations |url=https://ca-commercial.com/news/cambridge-analytica-and-scl-elections-commence-insolvency-proceedings-and-release-results-3 |website=CA Commercial |publisher=Cambridge Analytica |accessdate=2 May 2018 |language=en |date=2 May 2018 }}</ref> There are, as Cambridge Analytica claimed, other organizations (possibly including "{{w|cyberwarfare}}" arms of security services of different nation states) doing similar work, contributing to the Balkanization and exploitation of the international body politic. It is unclear how different Cambridge Analytica's activities differed from those of such other organizations that have so far not been similarly "vilified" and whether the demise of Cambridge Analytical will materially reduce the level of Balkanization created by these types of activities. Before leaving this discussion, it may be worth suggesting that "conservatism" as defined by the US corporate elite seems to have created the current environment of fact-free news and [[w:Post-truth politics|post-truth politics]]. The creation and maintenance of this environment seems to require denigrating fact checking. Three groups of people are generally more careful about checking their facts than the public at large: * Investigative journalists, * University professors, and * [[w:Wikipedia:Wikipedians|Wikipedians]]. All three have been under attack. The virtual elimination of investigative journalism from mainstream broadcasting in the US was discussed above. Beyond this, there have been numerous claims at least since the 1980s that the media have a liberal bias.<ref>For a discussion of this, see the Wikipedia article on [[w:Media bias in the United States|"media bias in the United States"]].</ref> Accusations of a [[w:Liberal bias in academia|liberal bias in academia]] date back at least to [[w: McCarthyism|Senator Joe McCarthy's]] infamous "second red scare," 1947-1956, when McCarthy and his followers made numerous "accusations of subversion or treason without proper regard for evidence." The Wikipedia article on "[[w:Liberal bias in academia|Liberal bias in academia]]" cites research by different people, with known conservatives and Libertarians working hard to document a perceived liberal bias, which other researchers failed to confirm. Liberal or conservative, most faculty members at major universities owe their positions to publications in refereed academic journals. Manuscripts submitted to serious academic journals must cite credible sources and otherwise provide solid documentation for what they say. [[w:Criticism of Wikipedia|Some conservatives have claimed that Wikipedia has a liberal bias.]]<ref>{{Citation | title = Examples of Bias in Wikipedia | publisher = Conservapedia | url = http://www.conservapedia.com/Examples_of_Bias_in_Wikipedia | accessdate = 2017-03-21}}</ref> In fact almost anyone can change almost anything on Wikipedia -- and almost anyone else can change (or even delete) what others wrote. What stays is generally written from a [[w:Wikipedia:Neutral point of view|neutral point of view]], citing [[w:Wikipedia:Verifiability|credible sources]].<ref>In a few cases, people editing Wikipedia from certain internet protocol (IP) addresses have been blocked, because of repeated attempts to burnish the images of some and attack opponents. See, e.g., [[w:United States Congressional staff edits to Wikipedia|United States Congressional staff edits to Wikipedia]].</ref> ==== 3.2.4. [[Media and corruption]] ==== [[Media and corruption|Econometric research has found that countries with greater press freedom tend to have less corruption.]]<ref>{{Citation | last = Brunetti | first = Aymo | last2 = Weder | first2 = Beatrice | author-link = w:de:Aymo Brunetti | author2-link = w:Beatrice Weder di Mauro | year = 2003 | title = A free press is bad news for corruption | journal = Journal of Public Economics | volume = 87 | publisher = Elsevier | pages = 1801-1824 | url = https://campus.fsu.edu/bbcswebdav/orgs/econ_office_org/Institutions_Reading_List/17._Corruption_and_Economic_Performance/Brunetti,_A._and_B._Weber-_A_Free_Press_is_Bad_News_for_Corruption | accessdate = 2017-06-24}}</ref> Greater political accountability and lower corruption were more likely where newspaper consumption was higher in data from roughly 100 countries and from different states in the US.<ref>{{Citation | last = Adserà | first = Alícia | last2 = Boix | first2 = Carles | author2-link = w:ca:Carles Boix i Serra | last3 = Payne | first3 = Mark | year = 2000 | title = Are You Being Served?: Political Accountability and Quality of Government | work = Working Paper | issue = 438 | publisher = Inter-American Development Bank Research Department | url = http://www.princeton.edu/~cboix/JLEO-paper.pdf | accessdate = 2014-08-17}} and {{Citation | last = Adserà | first = Alícia | last2 = Boix | first2 = Carles | author2-link = w:ca:Carles Boix i Serra | last3 = Payne | first3 = Mark | year = 2003 | title = Are You Being Served? Political Accountability and Quality of Government | journal = Journal of Law, Economics, & Organization | volume = 19 | publisher = Oxford U. Pr. | pages = 445-490 | url = http://www.princeton.edu/~cboix/JLEO-paper.pdf | accessdate = 2014-08-31}}</ref> A "poor fit between newspaper markets and political districts reduces press coverage of politics. ... Congressmen who are less covered by the local press work less for their constituencies: they are less likely to stand witness before congressional hearings ... . Federal spending is lower in areas where there is less press coverage of the local members of congress."<ref>{{Citation | last = Snyder | first = James M. | last2 = Strömberg | first2 = David | author2-link = w:sv:David Strömberg | year = 2008 | title = Press Coverage and Political Accountability | series = NBER Working Paper Series | issue = 13878 | publisher = National Bureau of Economic Research | url = http://www.nber.org/papers/w13878 | accessdate = 2014-08-17}}</ref> This was supported by an analysis of the consequences of the closure of the ''Cincinnati Post'' in 2007. The following year, "fewer candidates ran for municipal office in the Kentucky suburbs most reliant on the ''Post'', incumbents became more likely to win reelection, and voter turnout and campaign spending fell.<ref>{{Citation | last = Schulhofer-Wohl | first = Sam | last2 = Garrido | first2 =Miguel | year = 2009 | title = Do newspapers matter? Short-run and long-run evidence from the closure of the Cincinnati Post | series = NBER Working Paper Series | issue = 14817 | publisher = National Bureau of Economic Research | url = http://www.nber.org/papers/w14817 | accessdate = 2014-08-17}}</ref> An extreme example of media and corruption followed the closure around 1999 of the local newspaper in [[w:Bell, California|Bell, CA]], a city of roughly 35,000 near [[w:Los Angeles|Los Angeles]]. [[w:City of Bell scandal|The city manager, Robert Rizzo, decided, in essence, that the watchdog was dead, and it was time to party.]] He convinced other city leaders to join him. When the problems were documented in 2010, Rizzo's compensation was over $1 million per year, and the city was near bankruptcy. Rizzo was a "[[w:Control fraud|control fraud]]", in the parlance of [[w:William K. Black|Bill Black]], who was the lead litigator involved in sending [[w:Charles Keating|Charles Keating]] to prison in the [[w:Savings and loan crisis|Savings and loan scandal of the late 1980s and early 1990s]]. Black discussed this in his book, ''The Best Way to Rob a Bank is to Own One: How Corporate Executives and Politicians Looted the S&L Industry.''<ref>Black (2005)</ref> A "control fraud" is a leading executive who removes "the checks and balances on fraud within a company such as through the use of selective hiring and firing", especially of auditors. In 2014 [[w:Bill Moyers|Bill Moyers]] noted that, "No banking executives have been criminally prosecuted for their role in causing the biggest financial disaster since the Great Depression." Bill Black replied that, "Obama wouldn’t have been president but for the financial contribution of bankers.”<ref>{{Citation | last = Black | first = William K. | editor-last = Moyers | editor-first = Bill | date = October 3, 2014 | title = Too big to fail | publisher = Moyers & Company | url = http://billmoyers.com/episode/too-big-to-jail/ | accessdate = 2017-03-22}}</ref> One major conclusion of the present analysis is that campaign contributions like these would not likely have the impact they do in the US today if the public had more control over media content. From this perspective, it was easier for the mainstream media to expose Charles Keating than the banking executives who manufactured the [[w:Financial crisis of 2007–2008|Financial crisis of 2007–2008]] and manipulated the political process to benefit from it, because the advertising budgets of the control frauds of the Savings and loan industry were tiny relative to those of the current international bankers. [[File:Knowledge v. public media.png|thumb|Figure 10. Percent correct answers in surveys of knowledge of domestic and international politics vs. per capita subsidies for public media in the Denmark (DK), Finland (FI), the United Kingdom (UK) and the United States (US). Source: The "politicalKnowledge" data set in the "Ecdat" package available on the Comprehensive R Archive Network, based on research cited by McChesney and Nichols (2010).]] Even ignoring extreme cases like Bell, CA, the dumbing down of US commercial broadcasting was documented in research comparing the public's knowledge of current affairs between the US, the [[w:United Kingdom|United Kingdom (UK)]], [[w:Denmark|Denmark]] and [[w:Finland|Finland]]; see the accompanying Figure 10: College graduates in the US answered correctly roughly 70 percent of questions about political issues as people with the equivalent of high school in Denmark and Finland, while high school graduates in the US could only answer roughly 30 percent of the same questions. The primary difference was funding for mass media, according to McChesney and Nichols (2010): This was $1.35 per person in the US in 2007 vs. the equivalent of $101 in Denmark and Finland. The UK was in between: They spent the equivalent of $80 per person, and Brits with roughly 12th grade educations correctly answered almost 60 percent of the questions on average. The $101 per person per year invested in public media in Denmark and Finland seems comparable to the 0.2 percent of GDP that the US spent in citizen-directed subsidies under the [[w:Postal Service Act|US Postal Service Act of 1792]], discussed below in the section on "Media and nation building." On December 7, 2015, and February 29, 2016, [[w:Leslie Moonves|Les Moonves]], President and CEO of [[w:CBS Corporation|CBS]] bragged to investor conferences that the Trump campaign "may not be good for America, but it's damn good for CBS. ... The money's rolling in, this is fun."<ref>In 2012 he similarly said that "Super PACs may be bad for America, but they’re very good for CBS." {{Citation | last = Fang | first = Lee | date = February 29, 2016 | title = CBS CEO: “For Us, Economically, Donald’s Place in This Election Is a Good Thing” | journal = The Intercept | publisher = First Look Media | url = https://theintercept.com/2016/02/29/cbs-donald-trump/ | accessdate = 2017-03-22}}</ref> This was, in essence, an admission that CBS was sacrificing the best interests of the nation and humanity to favor the short term pecuniary interests of CBS. It seems virtually certain that all the other mainstream broadcasters in the US and Britain were doing essentially the same. The rules of business in the US almost certainly contributed not only to the Trump victory but to the US-led invasion of Iraq in 2003, the long-standing US support for the Saudi royal family, and the US role in destroying democracy in foreign countries, as discussed above in the section on "US foreign interventions in opposition to democracy." The Saudi connection, in turn, contributed to the [[w:September 11 attacks|September 11 attacks]] and the creation of ISIL, all discussed elsewhere in this essay. In particular, the role of the mainstream media in the US and the UK in US-led invasion of Iraq in 2003 might seem to qualify as "inciting a riot" under the standard of "[[w:Shouting fire in a crowded theater|falsely shouting fire in a crowded theater]]." Moonves and others in similar executive positions in the other mainstream broadcasters are safe, because the magnitude of the crime is too large for it to be widely understood, let alone prosecuted. In other words, these comments of CBS President Moonves combined with the other evidence summarized in this essay suggests that the current structure of the mainstream media in the US and Britain provide real and present threats to international peace and security.<ref>Similar things could be said about media in other countries. See, e.g., Peri (2012) for a similar analysis of the Israeli media.</ref> ==== 3.2.5. International business ownership of media ==== In France, the US and elsewhere, international business interests have ties to how the media are funded. One of the more obvious examples is [[w:Le Figaro|''Le Figaro'']], the oldest national daily newspaper in France and one of the most widely respected newspapers in the world. It is owned by the [[w:Dassault Group|Dassault Group]], one of the world’s leading arms merchants.<ref>Cagé (2016)</ref> ''Le Figaro'' therefore has a [[w:conflict of interest|conflict of interest]] in honestly reporting on anything that might question the advisability of any arms deal.<ref name="Feinstein2011">{{cite book|author=Andrew Feinstein |title=The Shadow World: Inside the Global Arms Trade|url=https://books.google.com/books?id=u06m3epox7wC|year=2011|publisher=Hamish Hamilton|isbn=978-0-241-14441-1}}</ref> It has a clear motive to overdramatize the problem of terrorism as long as they can do so in a way that supports further French weapons sales and more use of French overt and covert power to support repressive governments favored by French multinational executives, who advertise in ''Le Figaro''. Similarly, [[w:Westinghouse Electric Corporation|Westinghouse]] owned [[w:CBS Corporation|CBS]] between 1995 and 2000 before selling it to [[w:Viacom|Viacom]]. During that period, CBS and all the other mainstream commercial broadcasters in the US fired nearly all their investigative journalists, retaining only enough for popular shows like [[w:60 Minutes|''60 Minutes'']].<ref>{{Citation | last = McChesney | first = Robert W. | year = 2004 | title = The Problem of the Media | publisher = Monthly Review Press | page = 81 | isbn =1-58367-105-6}}</ref> Why? Because [[Media and corruption|investigative journalism threatens to expose questionable activities of people with substantive control over the media.]] * ''Investigative journalism is essential for democracy and a threat to people with substantial control over the media.'' As in France, the mainstream media in the US have a clear motive to overdramatize the problem of terrorism as long as they can do so in a way that supports further US sales of weapons, especially to [[w:List of authoritarian regimes supported by the United States|repressive governments favored by US multinational executives]]. ==== 3.2.6. How media personalities are selected ==== It seems reasonable to assume that most journalist believe they are honest, fair and balanced in what they report. The situation is similar to [[w:Lawrence Lessig|Lawrence Lessig]]’s description of [[w:Republic, Lost|the US Congress]]: Very few people in the US House and Senate demand and receive bribes. However, all are elected in a system that requires them to spend huge amounts of time asking people with lots of money for campaign contributions -- and then listening to their lobbyists to the exclusion of people who can not afford to buy such access. Candidates who are perceived to be unfriendly to the major campaign contributors (including especially those with control over major advertising budgets) are unlikely to get elected, as noted above. Similarly, media personalities are selected and fired depending on their ability to attract an audience while largely supporting the official line. [[w:Phil Donahue|Phil Donahue]] was fired from MSNBC in the runup to the 2003 US-led invasion of Iraq for trying to provide airtime to people who challenged the official rationale used to justify the invasion. He was dismissed in spite of having high ratings. [[w:Hutton Inquiry|BBC journalist Andrew Gilligan, chairman Gavyn Davies and director-general Greg Dyke resigned under fire]] for claiming that the British government had “sexed up” a report claiming Saddam Hussein had [[w:weapon of mass destruction|weapons of mass destruction (WMDs)]].<ref>[[w:Hutton Inquiry#Aftermath of publication|Davies resigned]] 2004-01-28, the day the [[w:Hutton Inquiry|Hutton Inquiry]] was published. Dyke resigned two days later, as did Gilligan. The Hutton Inquiry was a judicial inquiry in the UK chaired by Lord Hutton that opened 2003-08-01 to investigate the death 2003-07-18 of Dr David Kelly, a biological warfare expert and former UN weapons inspector in Iraq who had been named as the source for reports aired 2003-05-29 by Andrew Gilligan that the Blair government had "sexed up" the intelligence reports to justify British participation in the US-led invasion of Iraq 2003-03-20. See also <!-- Gilligan statement in full -->{{cite Q|Q111352244}}. </ref> The official [[w:Hutton Inquiry|Hutton Inquiry]] in 2003-2004 cleared the government of wrongdoing and strongly criticized the BBC. Thirteen years later on 6 July 2016, an official [[w:Iraq Inquiry|Iraq Inquiry]] of the British government acknowledged that the [[w:Blair ministry|Blair government]] had “sexed up” reports. Since the invasion it has become more widely known that Saddam Hussein had gotten chemical and biological warfare technology from the US, Britain and others with the support of Western governments<ref>{{Citation | date = 1990-09-11 | title = The arming of Iraq | periodical = Frontline | publisher = United States Public Broadcasting System (PBS) | url = http://www.pbs.org/wgbh/pages/frontline/shows/longroad/etc/arming.html | accessdate = 2017-02-26}}</ref> and [[w:Riegle Report|had used them against coalition forces]] in the [[w:Gulf War|1990-91 Gulf War]].<ref>{{Citation | title = Riegle Report | publisher = wikipedia | url = https://en.wikipedia.org/wiki/Riegle_Report | accessdate = 2017-02-26}}</ref> It was also reported that the US had invited three Iraqi nuclear scientists to the US in August 1989 for highly classified training on designing nuclear weapons.<ref>{{cite Q|Q106044626}}<!-- Building Saddam Hussein's bomb -->.</ref> [[w:Iraq and weapons of mass destruction|After the 2003 invasion relatively small numbers of weapons of mass destruction]] were found, but they all seemed old and degraded, thus substantiating the [[w:United Nations Special Commission|1999 comment from former UN weapons inspector Scott Ritter that Iraq had no militarily viable biological or chemical weapons on any meaningful scale.]]<ref>This quote came from an interview published on the web whose URL is no longer valid. A similar quote is available in {{cite web | last = Hurd | first = Nathaniel |date= 27 April 2000 |title=Interview with Scot Ritter |publisher= Campaign Against Sanctions on Iraq |url= http://www.fas.org/news/iraq/1999/07/990712-for.htm |accessdate= 2017-03-17}}</ref> Mainstream media executives in the US and Britain surely must have known that the official rationale for the 2003 US-led invasion of Iraq were questionable at best and possibly fraudulent. Apart from the BBC executives who resigned under fire for questioning the official rationale, they actively worked to suppress honest debate. In so doing they effectively stampeded the public in the US, Britain, and other countries in G. W. Bush’s “[[w:Coalition of the willing|Coalition of the willing]]” into supporting the invasion without adequate debate. In retrospect, it’s clear that the justification was at best wrong and likely fraudulent -- and the media executives should have known at the time that more evidence and debate were needed. === 3.3. Media and nation building === Robert McChesney and John Nichols claim the US has had three positive experience with nation building: its own and Germany and Japan after World War II. All three involved substantive subsidies for journalism.<ref>McChesney and Nichols (2010, esp. Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254)</ref> This section reviews publications discussing a possible relationship between media and nation building in the US, Germany, Japan, Iraq, Afghanistan, Brazil, and Africa. If you know of other evidence relevant to this question, please post a discussion of it here or on the companion "Discuss" page. ==== 3.3.1. United States ==== Under the [[w:Postal Service Act|US Postal Service Act of 1792]] newspapers were delivered up to 100 miles for a penny and beyond that for a penny and a half, when first class postage was between six and 25 cents depending on distance. This subsidy was citizen-directed and did not discriminate on content. The cost was roughly 0.2% of the economy (GDP, Gross Domestic Product), or just over $100 per person per year in today's money, according to McChesney and Nichols (2016).<ref>McChesney and Nichols (2016, p. 167) wrote, “If the United States government subsidized journalism in the second decade of the twenty-first century as a percentage of GDP to the same extent that it did in the first half of the nineteenth century, it would spend in the area of $35 billion annually.” The [[w:United States|US]] population in 2017 has been estimated at 325 million; $35 billion divided by 325 million is $108 per person. The US GDP for 2016 was reported to be $18.6 trillion; $35 billion divided by $18.6 trillion is $1.9 per thousand, which we round to 0.2 percent.</ref> That’s $2 per week for every man, woman and child in the US. ==== 3.3.2. Germany and Japan ==== US President [[w:Harry Truman|Harry Truman]] and his top military leaders including [[w:Dwight D. Eisenhower|Dwight Eisenhower]], and [[w:Douglas MacArthur|Douglas MacArthur]] were all veterans of World War I. They agreed that [[w:The war to end war|"the war to end war" (World War I)]] had not ended war, and they needed to do something different to prevent another war in another 20 years. To support the development of a democratic tradition, they forced the post-fascist governments in Germany and Japan to provide substantial subsidies for journalism. After the official German government surrendered, Eisenhower “called in German reporters and told them he wanted a free press. If he made decisions that they disagreed with, he wanted them to say so in print. The reporters having been under the Nazi regime since 1933, were astonished”.<ref>McChesney and Nichols (2010, Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254)</ref> Ike felt that the post-fascist media would lose credibility if they failed to criticize the occupiers. In addition, a cantankerous free press is an essential constraint on abuse of power by elites. ==== 3.3.3. Iraq ==== In discussing free press and media subsidies in the US and post-fascist Germany and Japan, McChesney and Nichols asked how the history of Iraq might be different if the US had made similar commitments to free press, “rather than [[w:Paul Bremer|L. Paul Bremer]]’s edicts -- which one Iraqi editor interpreted as, 'In other words, if you’re not with America, you’re with Saddam'".<ref>McChesney and Nichols (2010, p. 242)</ref> Might a free press in post-Saddam Iraq have dramatically limited the growth of the Islamic State? * ''Corruption grows to consume the available money.'' [[Media and corruption|The primary limit on political corruption is a free press]]. * ''[[q:Louis Brandeis|Sunlight is said to be the best of disinfectants]]''.<ref>This is from [[w:Louis Brandeis|Louis Brandeis]] (1914) ‘’[[w:Other People's Money and How Bankers Use It]]’’ (Frederick A. Stokes). Brandeis joined the US Supreme Court in 1916.</ref> Might a more vigorous press environment in both Iraq and the US have reduced the risk of a military disaster like that in Mosul in 2014? Sure. It might still have happened, but it would have been less likely. One symptom of conflicts of interest in both the mainstream media and the US congress is the fact that the [[w:Government Accountability Office investigations of the Department of Defense|Department of Defense (DoD) is the only US government agency to have failed every audit]] since all government agencies were required to pass such audits by the [[w:Chief Financial Officers Act|Chief Financial Officers Act of 1990]].<ref name=’GAO-DoD’>{{Citation | title = Government Accountability Office investigations of the Department of Defense | publisher = wikipedia | url = https://en.wikipedia.org/wiki/Government_Accountability_Office_investigations_of_the_Department_of_Defense | accessdate = 2017-02-27}}</ref><ref>{{cite web |last1=Smithberger |first1=Mandy |title=Will the Pentagon Ever Be Able to Be Audited? The Department of Defense remains the only federal agency that can’t get a clean audit opinion on its Statement of Budgetary Resources |url=http://www.pogo.org/straus/issues/defense-budget/2016/will-the-pentagon-ever-be.html |website=pogo.org |publisher=Project on Government Oversight |accessdate=2017-02-27 |date=March 28, 2016 }}</ref> ==== 3.3.4. Afghanistan ==== [[w:War in Afghanistan (2001–2014)|Might the US have invaded Afghanistan on October 7, 2001]], if the mainstream media had expressed more concern with the rule of law, including the request of the Afghani government for evidence of bin Laden's involvement in the suicide mass murders of September 11, 2001, before extraditing him?<ref name="theguardian.com"><!-- Guardian (2001-10-14) Bush rejects Taliban offer to hand Bin Laden over -->{{cite Q|Q111228506}}</ref> Might the mainstream media in the US and the UK have made more of an issue of this if multinational oil companies had less influence over government and media in both countries? These questions are impossible to answer with certainty, but one suspect that there would be less militarism and terrorism without these [[w:conflict of interest|conflicts of interest]]. How might Afghanistan be different today if the US had a more vigorous watchdog press, forcing US elected officials to require that the Department of Defense pass an audit, and Afghanistan and Iraq protect and subsidize a cantankerous press, as Truman, Eisenhower and MacArthur had done for Germany and Japan after World War II, as discussed above? A tentative answer to these questions appears in the 2015 book ''Thieves of State'' by [[w:Sarah Chayes|Sarah Chayes]]. She claimed that the Afghani government supported by the US has become a [[w:Kleptocracy|kleptocracy]] that collects bribes, not taxes, and reports people who do not pay appropriate bribes as [[w:Taliban|Taliban]] to the US-led forces there. US-led military units then kill the alleged Taliban. Chayes described multiple cases where corrupt Afghani officials were arrested for corruption then released after (she believes) intervention by the US [[w:Central Intelligence Agency|Central Intelligence Agency (CIA)]] to “protect their assets.”<ref>Chayes (2015). She says she went to Afghanistan with National Public Radio and stayed hoping to found a school for entrepreneurship. She found that the corruption made that effectively impossible. She shared her concerns with US military officers there, who recommended that their superiors listen to her. For a time she reported directly to [[w:Michael Mullen|Admiral Mike Mullen]], [[w:Chairman of the Joint Chiefs of Staff|Chairman of the Joint Chiefs of Staff (2007-2011)]]. She claims the CIA blocked her efforts to get that message to US President Obama.</ref> ==== 3.3.5. Brazil ==== "During the colonial period, Portugal made consistent efforts to reduce the economic, political and intellectual autonomy of Brazil," according to de Albuquerque.<ref name=deA>de Albuquerque (2012, p. 79)</ref> This condition improved slightly "when the Portuguese Court moved to Rio de Janiero," during the Napoleonic occupation of Portugal. However, "During the rest of the nineteenth century, most publications were leaflets, pamphlets and short-lived newspapers, dedicated chiefly to political polemics." Newspaper readership is still quite low compared to other countries: 60.6 per 1,000 adult population in 2000 vs. almost 12 times that in Norway (719.7) and 4.3 times that in the US (263.6).<ref name=deA/> [[w:History of Brazil|Brazil's current democracy dates from 1985.]] This Brazilian experience does not prove that better media help with nation building, but it is consistent with the claims of McChesney and Nichols (2010, 2016) discussed above. ==== 3.3.6. Africa ==== Cagé and Rueda studied newspaper readership and democratic engagement as a function of distance from Protestant missions with printing presses in Africa in 1903. Those missions printed educational material and public health information as well as the Christian [[w:Bible|Bible]] and related religious material. Cagé and Rueda found that, "within regions close to missions, proximity to a printing press is associated with higher newspaper readership, trust, education, and political participation" -- over a hundred years after the data on missions they used!<ref>{{Citation | last = Cagé | first = Julia | last2 = Rueda | first2 = Valeria | year = 2016 | title = The Long-Term Effects of the Printing Press in sub-Saharan Africa | journal = American Economic Journal: Applied Economics | volume = 8 | issue = 3 | pages = 69–99 | url = http://pubs.aeaweb.org/doi/pdfplus/10.1257/app.20140379 | accessdate = 2017-03-30}}</ref> Of course, empirical evidence is never complete.<ref>Quote from [[w:W. Edwards Deming|W. Edwards Deming]] from a public seminar in the 1980s.</ref> Still, this evidence is consistent with the general thrust of the other cases discussed in this section. === 3.4. Media funding and governance === Media is a [[w:Public good|public good]]. When elites control the editorial policies, it threatens democracy.<ref>See the discussion of the research by Kahneman (2011) elsewhere in this essay.</ref> We focus here on proposals to democratize funding and governance, focusing especially on the work of McChesney and Nichols (2010, 2021a, b) and Cagé (2016). ===== 3.4.1. Citizen-directed subsidies ===== McChesney and Nichols (2010, esp. ch. 4) discuss several different ways of providing democratically controlled subsidies for media. The [[w:Postal Service Act|US Postal Service Act of 1792]], discussed above, provides one example. News publications still get a modest postal subsidy in the US; McChesney and Nichols recommends increasing that, especially for publications with little or no advertising. They also suggest that the government could pay, e.g., up to half of journalists' salaries for publications with low circulation. “In exchange for accepting subsidies, post-corporate newspapers would be required to place everything they produce on the Web [in] the public domain -- creating vast new deposits of current and, ultimately, historical information.”<ref>McChesney and Nichols (2010, p. 189)</ref> They also propose a “Citizenship News Voucher”, whereby every American adult gets a $200 voucher that s/he can donate to any nonprofit news medium or combination of such nonprofits.<ref>McChesney and Nichols (2010, p. 201)</ref><ref>For more, see [[media and corruption]], especially regarding Bruce Ackerman’s proposal to distribute subsidies in proportion to qualified Internet clicks.</ref> McChesney and Nichols (2010, pp. 170-172) also recommend subsidizing high school newspapers and radio stations to help develop a democratic, civic culture in the youth. [[w:Bruce Ackerman|Bruce Ackerman]] proposed "Internet news vouchers" that ask Internet users to "click a box whenever they read a news article that contributes to their political understanding. ... [A] National Endowment for Journalism ... would compensate the news organization originating the article on the basis of a strict mathematical formula: the more clicks, the bigger the check from the Endowment."<ref>{{Cite book | last =Ackerman | first =Bruce | author-link = w:Bruce Ackerman | year =2010 | title =The Decline and Fall of the American Republic | chapter =5. Enlightening politics | publisher =Harvard U. Pr. | page =133 | isbn =978-0-674-05703-6}}</ref><ref>{{Citation | last =Ackerman | first =Bruce | author-link = w:Bruce Ackerman | date =May 6, 2013 | title =Reviving Democratic Citizenship? | journal =Politics & Society | volume =41 | issue =2 | publisher =Sage | pages =309–317 | url =http://pas.sagepub.com/content/41/2/309.abstract | accessdate =June 16, 2013 | doi=10.1177/0032329213483103}}</ref> Dan Hind proposed "public commissioning" of news, where "Journalists, academics and citizen researchers would post proposals for funding" investigative journalism on a particular issue with a public trust funded from taxes or license fees. "These proposals would be made available online and in print in municipal libraries and elsewhere. ... The public would then vote for the proposals it wanted to support."<ref>{{cite book |last= Hind |first= Dan |title= The Return of the Public |year= 2010 |publisher= Verso | pages = 159-160 | chapter = 10. Public Commissioning | isbn=978-1-84467-594-4}}</ref> [[w:Dean Baker|Dean Baker]] suggested an "Artistic Freedom Voucher," similar to these other options but not limited to journalism: He claims that the copyright system today, at least in the US, locks up entirely too much information behind paywalls.<ref name=Baker2003>{{Citation | last = Baker | first = Dean | date = November 5, 2003 | title = The Artistic Freedom Voucher: An Internet Age Alternative to Copyrights | type = Briefing paper | publisher = Center for Economic and Policy Research | url = http://cepr.net/publications/reports/the-artistic-freedom-voucher-internet-age-alternative-to-copyrights | accessdate = 2017-03-30}}</ref><ref>See also [[w:Free Culture (book)|''Free Culture'']]: {{cite book |title= Free Culture: How Big Media Uses Technology and the Law to Lock Down Culture and Control Creativity | edition=US paperback |isbn=978-82-690182-0-2 |author=Lessig, Lawrence | publisher=Petter Reinholdtsen |date=2015 }}</ref> His idea, therefore, was to provide citizen-directed subsidies for virtually any artistic creation that would be placed in the public domain -- on the web to the extent that it can be digitized. This would make it easier for aspiring artists, performers, or writers to get started. After they become well enough known, they could stop accepting "Artistic Freedom Voucher" money and sell what they produce using the existing copyright system.<ref name=Baker2003/> This may be used if it seems too difficult to develop an acceptable legal definition of "investigative journalism;" see the discussion of this in section "4.4.3. Citizen-directed subsidies" for media below. What's the optimal level of funding for investigative journalism? The best information available, at least in this essay, is the 0.2 percent of GDP, suggested by McChesney and Nichols (2010), and the comparable amount in Scandinavia, discussed with Figure 10 above. More recently McChesney and Nichols (2021a, b) propose a Local Journalism Initiative (LJI), whereby the US would be divided into local jurisdictions, typically counties, and each would be given 0.15 percent of Gross Domestic Product (GDP), currently just over $100 per person, that would be distributed to local nonprofit journalism organizations based on regular elections where each adult would have three votes to select their three most preferred qualifying organizations. Counties with low populations might be combined as might counties in a metropolitan area. And residents in areas with large populations may get four or five votes rather that three "to guarantee diversity of voices." To qualify, the recipient organization could not be a subsidiary of a larger organization and would have to devote 75 percent of their salaries to journalists based in that local jurisdiction. Each such organization would have to produce and publish original material at least five days per week on its website; no restrictions should be placed on the content. Roughly that amount of money might be obtained without action by the federal government in local jurisdictions that chose to match what they spend on [[Confirmation bias and conflict#Advertising and accounting|accounting, advertising, media and public relations]] with these kinds of citizen-directed subsidies for journalism. The discussions above of terrorism, conflict, economics, and incarcerations including the [[w:City of Bell scandal|scandal in the city of Bell, CA]], and "control frauds" suggests that it might be wise to provide citizen-directed subsidies for investigative journalism comparable to what organizations spend on [[w: Accounting|accounting]]; see "Implications" for "Subnational" entities below for more on this. ==== 3.4.2. Nonprofit media organization (NMO) ==== [[w:Julia Cagé|Julia Cagé]] proposes a new model of funding and governance for media she calls a ‘’Nonprofit media organization’’. This is a charitable foundation with democratic governance split between the funders, the journalists, and their audience.<ref>Cagé (2016)</ref> ==== 3.4.3. Net neutrality ==== The fight over [[w:Net neutrality|net neutrality]] is a question of how media will be funded and governed. [[w:Political positions of Donald Trump#Technology and net neutrality|US President Donald Trump is opposed to net neutrality]]. In September 2015 he was quoted as saying, "Obama's attack on the Internet is another top down power grab. Net neutrality is the [[w:Fairness Doctrine|Fairness Doctrine]]. Will target the conservative media."<ref>Caroline Craig, [http://www.infoworld.com/article/2986220/net-neutrality/where-the-candidates-stand-on-net-neutrality.html Where the candidates stand on Net neutrality], ''InfoWorld'' (September 25, 2015).</ref> [[w:Ajit Varadaraj Pai|Ajit Pai]], Trump's Chairman of the US [[w:Federal Communications Commission|Federal Communications Commission (FCC)]], agrees: He has claimed that [[w:Ajit Varadaraj Pai#First Amendment issues|net neutrality]] in an attempt to weaken the "culture of the First Amendment,"<ref>{{Cite news |url=http://www.washingtonexaminer.com/fcc-commissioner-something-changing-in-america-about-the-first-amendment/article/2585434 |title=FCC commissioner: Something changing in America about the First Amendment |newspaper=Washington Examiner |first=Rudy |last=Takala |date=March 14, 2016 }}</ref> because it deprives [[w:Internet service provider|Internet service providers (ISPs)]] of their freedom of speech. He said it was "conceivable" that the FCC would seek to regulate political speech offered by edge providers such as [[w:Fox News|Fox News]] or the [[w:Drudge Report|Drudge Report]].<ref>{{Cite news |url=http://www.washingtonexaminer.com/fcc-commissioner-u.s.-tradition-of-free-expression-slipping-away/article/2583354 |title=FCC commissioner: U.S. tradition of free expression slipping away |newspaper=Washington Examiner |first=Rudy |last=Takala |date=February 16, 2016 }}</ref> Pai is correct that net neutrality limits the free speech rights of ISPs. One implication of this essay is that there is a compelling national security interest in doing something to improve the way in which news is selected, produced, supplied to and consumed by the public. Net neutrality is one such measure that has broad support among the US public.{{Citation needed|reason=document the popularity|date=March 2017}} Limits on free speech appear in the [[w:Hatch Act of 1939|Hatch Act]] and Department of Defense Directive 1344.10, which prohibit employees of the US executive branch from engaging in certain types of political activities. Whether ISPs are allowed to provide content or not, [[w:Net neutrality|net neutrality]] could be improved by a tax on advertising placed with content-providing [[w:Internet service provider|Internet service providers]]. ISPs that were common carriers would not have to pay this tax, nor would content providers that were not ISPs. == 4. Implications == The above discussion suggests policy implications at five levels: personal, interpersonal, subnational, national, and international. * ''The primary difference between rich and poor countries is politics.'' * ''The primary difference between rich and poor people within a country is politics.'' === 4.1. Personal === Individuals need to understand that unless they control the funding for the media they patronize, changes in their behaviors are being sold to the people who do control the funding. The discussion above illustrates only a few of the problems this generates. ==== 4.1.1. Fact checking ==== The research of Daniel Kahneman (2011) indicate that humans have two methods for ascertaining truth: # How does it fit with my preconceptions? # How does it fit with credible sources? The first approach leads to many decision errors, including blind support by the American public for the destruction of democracy in Iran, Guatemala, Brazil, and Chile, the cancelation of elections in Cuba and Vietnam, all without public debate, and the invasion of Iraq with severely restricted debate, as discussed above. This suggests that society would benefit if a critical mass of the electorate were to actively search for more credible sources of information on the most important issues of the day and then discuss what they learn with others. The good news for this is Chenoweth's 3.5 percent rule: Of the 323 major governmental change efforts of the twentieth century (summarized in Tables 1 and 3, Figure 6, and Appendix 2), every one that got the active support of at least 3.5 percent of the population was successful -- and all of those were nonviolent.<ref>{{Citation | last = Chenoweth | first = Erica | date = November 4, 2013 | title = My Talk at TEDxBoulder: Civil Resistance and the “3.5% Rule” | publisher = The Rational Insurgent | url = https://rationalinsurgent.com/2013/11/04/my-talk-at-tedxboulder-civil-resistance-and-the-3-5-rule/ | accessdate = 2017-03-21}}</ref><ref>{{Citation | last = Chenoweth | first = Erica | date = 2017-02-01 | title = It may only take 3.5% of the population to topple a dictator – with civil resistance | publisher = The Guardian | url = https://www.theguardian.com/commentisfree/2017/feb/01/worried-american-democracy-study-activist-techniques | accessdate = 2017-03-21}}</ref> The [[w:Indivisible movement|Indivisible movement]] in the US may already have that many supporters. However, for them to effect substantive change beyond blunting the agenda of President Trump, they may need to check their facts more carefully. Otherwise, they may succeed in blocking the worst parts of President Trump's agenda but fail to substantively alter the continued transfer of wealth from the poor and middle class to "job creators," summarized in Figure 7 above. ==== 4.1.2. Turn off the mainstream media ==== Turn off the mainstream media. Support instead noncommercial investigative journalism with transparent funding that places everything they produce on the web in the public domain, as suggested in the discussion of media funding and governance above. Good nonprofit media organizations are not always easy to find. One reasonable list of suggestions is available from {{Citation | date = February 6, 2017 | title = How to Find and Support Trustworthy Journalism | publisher = DailyGood | url = http://www.dailygood.org/story/1505/how-to-find-and-support-trustworthy-journalism-democracy-fund/ | access-date = 2017-03-31 }}. Two not listed there are the following: * [[w:Democracy Now!|''Democracy Now!,'']] which produces a one-hour daily news broadcast, Monday through Friday, that “is funded entirely through contributions from listeners, viewers, and foundations and does not accept advertisers, corporate underwriting or government funding.” They also post transcripts on their web site. * AllSides.com,<ref>{{Citation | title = AllSides | publisher = AllSides.com | url = https://www.allsides.com/ | accessdate = 2017-03-29}}</ref> which provides side-by-side comparison of how typically left, center and right sources cover a particular story. Interested readers are invited to add to this list any nonprofit news organization not listed here or in the "How to ..." site mentioned, especially if they have transparent funding and makes everything they produce available on the web in the public domain (or a relatively unrestricted license like the [[w:Creative Commons license|Creative Commons Attribution ShareAlike license]]). ==== 4.1.3. Make politics a primary entertainment. ==== The discussion of Kahneman's work above suggests we should not assume that current knowledge is adequate. No human can possibly check their facts on everything. That's part of why we make so many decisions based on what comes most readily to mind. However, Kahneman says we would better ourselves and others if we identified a few very important issues and spend time and money looking for alternative sources of information on those issues. This should include looking for information that might conflict with our preconceptions. How can we get time for this? Turn off the mainstream media, as suggested in the previous section, and make the search for alternative information a primary entertainment. === 4.2. Interpersonal === People say, “We don’t talk politics.” In a democracy that’s undemocratic. If we don’t talk politics, it becomes easier for corrupt elites to divide and conquer the poor and the middle class, getting them to support policies that benefit the elites at the expense of everyone else. Examples include the public support for the War on Terror and the “get tough on crime” wave that drove the five-fold increase in the incarceration rate described above. In both these examples, the results appear to have been detrimental to society as a whole, to the extent that the analysis above is accurate. We need to talk politics. We should not argue. Instead, we need to ask questions, listen with respect and show our audience that we’ve heard their concerns before we share our perspective.{{Citation needed|reason=This process has a name and supporting literature, but I forget right now what it is.|date=March 2017}} * ''We should strive to agree to disagree agreeably.'' Also, people with computer skills can help others improve their ability to use computers to get better information. === 4.3. Subnational === As suggested above, governmental organizations at all levels, including subnational entities (e.g., state and local governmental organizations in the US) might benefit their constituents by devoting a portion of their budget roughly comparable to what they spend on [[w:Accounting|accounting]] to something like an Endowment for Journalism that would provide citizen-directed subsidies for local investigative journalism organizations. A system like this might also be funded in part by businesses and ordinary citizens, who would like to subsidize investigative journalism in their service area or community. More experiments like this are needed. === 4.4. National === For national reforms, at least in the US, the above discussion favors [[w:Net neutrality|net neutrality]] and major limits on government secrecy. ==== 4.4.1. Net neutrality ==== As noted above, [[w:Political positions of Donald Trump|President Trump]] opposes [[w:Net neutrality|net neutrality]]<ref>{{Citation | last = Craig | first = Caroline | date = September 25, 2015 | title = Where the candidates stand on Net neutrality | journal = InfoWorld | url = http://www.infoworld.com/article/2986220/net-neutrality/where-the-candidates-stand-on-net-neutrality.html | accessdate = 2017-03-09}}</ref> and supports Internet censorship.<ref>{{Citation | last = Thielman | first = Sam | date = March 14, 2016 | title = Tech policy activists find Bernie Sanders is best bet – while Trump is the worst | newspaper = Guardian | url = https://www.theguardian.com/us-news/2016/mar/14/election-2016-tech-policy-net-neutrality-bernie-sanders-donald-trump | accessdate = 2017-03-09}}</ref> Internet censorship would put government bureaucrats in charge of fact checking. Destroying net neutrality would make it harder for consumers to obtain information that is not subsidized by big money interests. This in turn supports a continuation of the biased reporting that created the five-fold increase in incarcerations in the US, discussed above, and the "[[w:935 Lies|935 lies]]"<ref>Lewis (2014)</ref> that stampeded the Western world into invading Iraq on fraudulent grounds in 2003. If net neutrality is destroyed, congress ''could'' later overturn that action, though precedents for that are not encouraging. Congress could, however, encourage common carriers by taxing ISPs that also provided content, as suggested above. ==== 4.4.2. Limiting government secrecy ==== The 1995 [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] in the US made a number of recommendations including the following: * Excessive secrecy has significant consequences for the national interest when policy makers are not fully informed, the government is not held accountable for its actions, and the public cannot engage in informed debate. * Some secrecy is important to minimize inappropriate diffusion of details of weapon systems design and ongoing security operations as well as to allow public servants to secretly consider a variety of policy options without fear of criticism. * The best way to ensure that secrecy is respected, and that the most important secrets remain secret, is for secrecy to be returned to its limited but necessary role. Secrets can be protected more effectively if secrecy is reduced overall. Politicized intelligence was one contributor to the US-led invasion of Iraq in 2003, without which ISIL might never have been big enough to make international headlines, if it existed at all. This suggests two additional reforms: * Make all the intelligence services report to the US [[w:Government Accountability Office|Government Accountability Office (GAO)]]. * Drastically limit the duration of secrecy of any classified information created by intelligence services. The duration should be long enough to protect the element of surprise in democratically authorized military operations but short enough to make it much more difficult for the US government to interfere in the internal affairs of foreign countries without a full and open public debate.<ref>Information allowing the identification of specific individuals involved in questionable activities prior to the passage of reform legislation suggested herein might be classified for 30 years to protect those individuals. Such protections should also extend to private considerations of options considered but not implemented, to allow public officials to consider all options without fear of retaliation for options not submitted for public consideration. However, such protection could not extend to protect information from challenge in legal proceedings under US v. Reynolds, because US v. Reynolds seems to be a threat to US national security, and should be overturned, according to the argument in this section and supporting evidence.</ref> Making the intelligence services report to the GAO would not eliminate politicization of intelligence,<ref>[[w:Government Accountability Office investigations of the Department of Defense|As noted above, the US DoD failed to pass an audit since first required to do so in 1990.]] This is partly a result of interference by elected representatives.</ref> but it would limit the ability of the executive branch to do it on its own initiative, as seems to have been the case in the run-up to the US-led invasion of Iraq in 2003.<ref>as documented, e.g., by Lewis (2014)</ref> The Moynihan Commission recommended "returning secrecy to its limited but necessary role." Limiting the duration of classification to something like six months could be part of that. Six months seems long enough that it would not likely seriously impede any active military operations but short enough to effectively eliminate US efforts to destabilize foreign governments without a full and open public debate. We do not need ''uninformed'' "debates," like those that stampeded the US congress into approving the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964 or the use of force in Iraq in 2003. Like [[w:Truth and reconciliation commission|truth and reconciliation processes]], military personnel and other government employees should be given wide latitude for honest mistakes. However, the history of previous US government efforts to destroy the prospects for democracy in foreign countries suggests a need for a substantially shorter period of classification than is the practice today. To these and Moynihan's reforms, we would add one more: * Overturn the US Supreme Court decision in [[w:United States v. Reynolds|US v. Reynolds]]. Under that decision, no judge and no defendant can question the government’s claim of national security. US v. Reynolds has effectively given US government officials who can classify a document the ability to conceal malfeasance and criminal activities. For example, in the trial of [[w:Daniel Ellsberg|Daniel Ellsberg]] for leaking the [[w:Pentagon Papers|Pentagon Papers]], Ellsberg was not allowed to argue that the information he released was improperly classified. Recently, Ellsberg said that [[w:Edward Snowden|Ed Snowden]] could not get a fair trial in the US;<ref>{{Citation | last = Ellsberg | first = Daniel | date = 2014-05-30 | title = Snowden would not get a fair trial – and Kerry is wrong | newspaper = The Guardian | url = https://www.theguardian.com/commentisfree/2014/may/30/daniel-ellsberg-snowden-fair-trial-kerry-espionage-act | accessdate = 2017-03-09}}</ref> US v. Reynolds effectively says that a fair trial is impossible in any civil or criminal case involving US [[w:Classified information|information that could plausibly be classified]]. One more example: Wikileaks recently disclosed that the CIA has the ability to hack many devices including modern automobiles.<ref>{{Citation | date = 2017-03-07 | title = Vault 7: CIA Hacking Tools Revealed | publisher = WikiLeaks | url = https://wikileaks.org/ciav7p1/ | accessdate = 2017-03-14}}</ref> The Washington Post noted, "The fear that your car can be hacked and made to crash is not new, and it’s not completely unfounded. ... Concerns about automotive cyber security have been raised since automakers began outfitting cars and trucks with computer-controlled systems. ... [S]atellite, Bluetooth and Internet ... make them more vulnerable to hackers who can then gain access to the computerized systems without ever stepping foot near the actual vehicle. ... The WikiLeaks release even renewed suspicions about the death of journalist [[w:Michael Hastings|Michael Hastings]], who was killed in a single-car accident in Los Angeles in 2013."<ref>{{Citation | last = Overly | first = Steven | date = 2017-03-08 | title = What we know about car hacking, the CIA and those WikiLeaks claims | newspaper = Washington Post | url = https://www.washingtonpost.com/news/innovations/wp/2017/03/08/what-we-know-about-car-hacking-the-cia-and-those-wikileaks-claims/ | accessdate = 2017-03-14}} </ref> It would be irresponsible to say that the CIA killed Hastings. However, given the CIA's history briefly summarized above, it would be equally irresponsible to claim that they did not have the means and a history of far worse. Moreover, Hastings told others before his death that he was working on a story involving the CIA. If that's true, CIA personnel likely knew. This would have given them a motive, especially since an earlier article by Hastings forced the resignation of General [[w:Stanley McChrystal|Stanley McChrystal]]. The recommendations of the Moynihan Commission have so far been ignored. Why? The discussion above suggests that major US international business executives likely believe they benefit from having the US government promote regime change in foreign countries on their behalf. Since many of those international businesses also control major advertising budgets, the mainstream commercial media as currently structured have a conflict of interest in honestly reporting on those activities. This is very clear in some cases, less clear in others. Big oil, for example, seems to have benefitted from the US support for the Saudis since the 1930s. They also seem to have benefitted from the [[w:March 1949 Syrian coup d'état|destruction of democracy in Syria in 1949]] and [[w:1953 Iranian coup d'état|Iran in 1953]] as well as from the [[w:War in Afghanistan (2001–2014)|US-led invasion of Afghanistan in 2001]], as noted above. For another example, consider [[w:Allen Dulles|Allen Dulles]]: He was head of the CIA during the [[w:1954 Guatemalan coup d'état|1954 Guatemalan coup d'état]]. He sat on the board of directors of [[w:United Fruit Company|United Fruit]], a major beneficiary of the coup. ==== 4.4.3. Research on why people support one side or another in conflict and change their support over time ==== The world needs an "International Conflict Observatory" doing research that can not be kept secret into what motivates people to leave the sidelines to support one side or the other in conflict, to increase or decrease their level of support over time, and to desert or defect, when they do. The results of such research should be freely available, in the {{w|public domain}}, produced in a way that would not allow any government to try to classify it to keep it from the public, as discussed in the section above on [[Winning the War on Terror#2.12. G. W. Bush: "Why do they hate us?"|G. W. Bush: "Why do they hate us?"]]. ==== 4.4.4. Citizen-directed subsidies for media ==== The discussion above suggests a need for citizen-directed subsidies for investigative journalism on the order of 0.2 percent of GDP with a preference for non-commercial investigative journalism with transparent funding that puts everything they produce on the web in the public domain.<ref>See also {{Citation | title = Endowment for Journalism | publisher = Endowment for Journalism | url = http://endowmentforjournalism.org | accessdate = 2022-09-04}}</ref> More research is needed into whether and how "investigative journalism" might be defined so politicians could not easily divert these subsidies to organizations that supported only their political agenda. If this seems too difficult, some or all of the money given to an "Endowment for Journalism" that manages these funds could be disbursed under citizen direction to virtually all creative artists and / or nonprofit media organizations with transparent funding, who agree to place all they produce on the web in the public domain in exchange for citizen-directed subsidies; see also section "3.4.1. Citizen-directed subsidies" above. As noted above, McChesney and Nichols suggested that each taxpayer be given a tax rebate of up to $200 that they can split between qualifying nonprofit media organizations. An alternative might be to provide, e.g, five to one matches for small dollar amounts given to qualifying nonprofits up to a maximum of, e.g, $50 for every man, woman and child (max subsidy = $250 for each).<ref>The matching funds for minors could be "with the approval of their parents / legal guardians."</ref> Consistent with the recommendations of Cagé, outlined above, it may be wise to require the recipients to be "Nonprofit Media Organizations" (NMOs) meeting her requirements. An alternative might involve an "Endowment for Journalism" that would distribute funds in proportion to qualified Internet clicks. A system like this could be used by a local governmental entity, subsidizing only selections made by residents in that jurisdiction, or even a business wanting to promote local transparency in government within their primary service area.<ref>Qualifications would apply both to the individual making the click and the recipient organization. They would ensure that both the individual and the recipient would have an appropriate connection to the indicated governmental entity or business.</ref> === 4.5. International === The discussion above suggests several actions that can be taken by any country in the world. # Strengthen international law. # Support further research into the long-term impact of alternative approaches to conflict (effective defense). # Support research and dissemination of information on what motivates people on all sides of violent conflict to do what they do. # Support the widespread dissemination of the research into the relative effectiveness of violence and nonviolence and techniques of nonviolent civil disobedience. # Support free press everywhere. This includes increasing protections for journalists both domestically and by placing a national security tax on trade with countries with a documented history of mistreating journalists. It also includes supporting citizen-directed subsidies for noncommercial investigative journalism -- including in foreign countries. In particular, the security of Western nations could be enhanced through financial support for citizen-directed investigative journalism in foreign countries that also encouraged nonviolent civil disobedience in seeking redress of grievances. # Support research and experimentation with demand-side economics, as mentioned above. # Non-nuclear nations could place a “national defense tax” on trade with nuclear states to encourage nuclear disarmament. # Limit arms trade only to functioning democracies that vigorously support free press everywhere. # Accept refugees and support their adjustment with demand-side economics, as discussed above. # Drastically limit the use of airpower, including armed drones, only to support of ground operations. == 5. Summary == The above discussion summarizes research suggesting that the approach that has been taken so far to combating international terrorism has increased, not decreased, that threat. People everywhere can help reduce the risks described herein by being more selective in the media they choose to consume and by asking more questions. Research by Kahneman (2011), outlined above, make it clear that virtually all conflicts are driven in part by things people think they know that aren't so. People everywhere can also learn more about nonviolent civil disobedience. You can build civil society by listening respectfully to others including those with whom you may disagree, asking questions, and summarizing what you think you heard to show them that they've been heard. Then the others may be more willing to listen to your concerns. This may not apply everywhere, because local law and other considerations may make it too risky to discuss certain issues openly. The “Implications” section of this essay also includes suggestions for changes in national and international policies. == See also == The questions raised in this article are almost benign relative to the issues raised in the two more recent pieces: * [[1998 Embassy bombings and September 11]] * [[Expertise of military leaders and national security experts]] == References == * {{Citation | last = Bapat | first = Navin A. | year = 2011 | title = Transnational terrorism, US military aid, and the incentive to misrepresent | journal = Journal of Peace Research | volume = 48 | issue = 3 | pages = 303-318}} * {{Citation | last = Black | first = William K. | year = 2005 | title = The Best Way to Rob a Bank is to Own One: How Corporate Executives and Politicians Looted the S&L Industry | publisher = U. of Texas Pr. | isbn = 0-292-72139-0}} * {{Citation | last = Cagé | first = Julia | year = 2016 | origyear = 2015 | title = Saving the media: Capitalism, crowdfunding and democracy | publisher = Belknap | isbn = 9780674659759}} * {{Citation | last = Cassidy | first = John | date = November 18, 2015 | title = The Economics of Syrian Refugees | magazine = The New Yorker | url = http://www.newyorker.com/news/john-cassidy/the-economics-of-syrian-refugees | accessdate = 2017-03-13}} * {{Citation | last = Chayes | first = Sarah | year = 2014 | title = Thieves of State: Why Corruption Threatens Global Security | publisher = Norton | isbn = 978-0393239461}} * {{Citation | last = Chenoweth | first = Erica | last2 = Stephan | first2 = Maria J. | author-link = w:Erica Chenoweth | year = 2011 | title = Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict | publisher = Columbia U. Pr. | isbn = 978-0-231-15683-7}}. For their data see, {{Citation | last = Chenoweth | first = Erica | title = NAVCO Data Project | publisher = Sié Chéou-Kang Center for International Security & Diplomacy, Josef Korbel School of International Studies, University of Denver | url = http://www.du.edu/korbel/sie/research/chenow_navco_data.html | accessdate = 2017-03-17}} * {{Citation | last = Choi | first = Seung-Whan | last2 = James | first2 = Patrick | year = 2016 | title = Why does the United States intervene abroad? Democracy, human rights, and terrorism | journal = Journal of Conflict Resolution | volume = 60 | issue = 5 | publisher = Sage | pages = 899-926}} * {{Citation | last = de Albuquerque | first = Afonso | editor-last = Hallin | editor-first = Daniel C. | editor2-last = Mancini | editor2-first = Paolo | year = 2012 | title = Comparing media systems beyond the Western world | chapter = 5. On models and margins: Comparative media models viewed from a Brazilian perspective | publisher = Cambridge U. Pr. | pages = 72-95 | isbn = 978-1-107-69954-0}} * {{Citation | last = Focke | first = Florens | last2 = Niessen-Ruenzi | first2 = Alexandra | last3 = Ruenzi | first3 = Stefan | date = 2016-03-03 | title = A friendly turn: Advertising bias in the news media | publisher = SSRN | url = https://ssrn.com/abstract=2741613 | accessdate = 2017-05-11}} * {{Citation | last = Frum | first = David | last2 = Perle | first2 = Richard | year = 2004 | title = An end to evil: How to win the War on Terror | publisher = Ballantine | isbn = 0-345-47717-0}} * {{Citation | last = Graves | first = Spencer | date = 2005-02-26 | origyear = 2004 | title = The Impact of Violent and Nonviolent Action on Constructed Realities and Conflict | publisher = Productive Systems Engineering | url = http://prodsyse.com/conflict/Nonviolence&Reality.pdf | accessdate = 2017-02-20}} * {{Citation | last = Graves | first = Spencer | year = 2017 | chapter = terrorism | title = Ecdat | edition = 0.3-2 | publisher = Comprehensive R Archive Network (CRAN) and R-Forge | url = https://cloud.r-project.org/web/packages/Ecdat/index.html | accessdate = 2017-02-17}} * {{Citation | editor-last = Hallin | editor-first = Daniel C. | editor2-last = Mancini | editor2-first = Paolo | year = 2012 | title = Comparing media systems beyond the Western world | publisher = Cambridge U. Pr. | isbn = 978-1-107-69954-0}} * {{Citation | last = Hanushek | first = Eric A. | last2 = Peterson | first2 = Paul E. | last3 = Woessmann | first3 = Ludger | year = 2013 | title = Endangering Propsperity: A global view of the American school | publisher = Brookings Institution Pr. | isbn = 978-0-8157-0373-0}} * {{Citation | last = Hanushek | first = Eric A. | last2 = Woessmann | first2 = Ludger | year = 2015 | title = The knowledge capital of nations | publisher = CESifo | isbn = 978-0-262-02917-9}} * {{Citation | last = Harris | first = Brayton | year = 1999 | title = Blue & gray in black & white : newspapers in the Civil War | publisher = Brassey's | isbn = 978-1453617021}} * {{Citation | last =Horowitz | first =Michael | last2 =Reiter | first2 =Dan | year =2001 | title =When does aerial bombing work? Quantitative empirical tests, 1917-1999 | journal =Journal of Conflict Resolution | volume =45 | issue =2 | pages = 147–-173 | url =http://jcr.sagepub.com/content/45/2/147 | accessdate =July 29, 2013 | doi=10.1177/0022002701045002001}} * {{cite Q|Q57515305}}<!-- Jones and Libicki (2008) How terrorist groups end--> * {{Citation | last = Kahneman | first = Daniel | year = 2011 | title = Thinking, Fast and Slow | publisher = Farrar, Straus and Giroux | isbn = 978-0374275631}} * {{Citation | last = Lewis | first = Charles | author-link = w:Charles Lewis (journalist) | year = 2014 | title = [[w:935 Lies: The future of truth and the decline of America’s moral integrity|935 Lies: The future of truth and the decline of America’s moral integrity]] | publisher = Public Affairs | isbn = 978-1-61039-117-7}} * {{Citation | last = Lloyd | first = Mark | last2 = Friedland | first2 = Lewis A. | year = 2016 | title = The Communications Crisis in America, And How to Fix It | publisher = Palgrave Macmillan | isbn = 1-349-95030-0}} * {{cite Q|Q104888067}}<!-- McChesney and Nichols (2010) The Death and Life of American Journalism-->. * {{Citation | last = McChesney | first = Robert W. | last2 = Nichols | first2 = John | author-link = w:Robert W. McChesney | author2-link = w:John Nichols (journalist) | year = 2016 | title = People get ready: The fight against a jobless economy and a citizenless democracy| publisher = Nation Books | isbn = 9781568585215}} * <!-- The Local Journalism Initiative: a proposal to protect and extend democracy -->{{cite Q|Q109978060|date=2021a}} * <!-- To Protect and Extend Democracy, Recreate Local News Media -->{{cite Q|Q109978337|date=2021b}} * <!-- Pape (1996) Bombing to win: air power and coercion in war -->{{cite Q|Q107458786}} * {{cite Q|Q5318649}}<!--Pape 2005 Dying to win--> * {{Citation | last = Pape | first = Robert | last2 = Feldman | first2 = James K. | author-link = w:Robert Pape | year = 2010 | title = Cutting the fuse : the explosion of global suicide terrorism and how to stop it | publisher = U. of Chicago Pr. | isbn = 9780226645605}} * {{Citation | last = Peri | first = Yoram | editor-last = Hallin | editor-first = Daniel C. | editor2-last = Mancini | editor2-first = Paolo | year = 2012 | title = Comparing media systems beyond the Western world | chapter = 2. The impact of national security on the development of media systems: The case of Israel | publisher = Cambridge U. Pr. | pages = 11-25 | isbn = 978-1-107-69954-0}} * {{cite book|last1=Sacco|first1=Vincent F|title=When Crime Waves|date=2005|publisher=Sage|isbn=0761927832}} and {{cite book|last1=Youngblood|first1=Steven|title=Peace Journalism Principles and Practices|date=2017|publisher=Routledge|isbn=978-1-138-12467-7|pages=115-131}} * {{cite book |last1= Tyler |first1= Tom R. |last2= Huo |first2= Yuen J. |title= Trust in the Law: Encouraging Public Cooperation with the Police and Courts |year= 2002 |publisher= Russell Sage Foundation |isbn= 0871548895}} == Appendix 1. Terrorism death trends for the dozen countries with the most terrorism deaths, 2014-2015 == Figures 1 and 2 above and the plots in this Appendix were all created from the [[w:Global Terrorism Database|Global Terrorism Database]] using the summaries in Graves (2017). [[File:Terrorism deaths in Iraq.svg|thumb|Terrorism deaths in Iraq, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Nigeria.svg|thumb|Terrorism deaths in Nigeria, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Afghanistan.svg|thumb|Terrorism deaths in Afghanistan, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Syria.svg|thumb|Terrorism deaths in Syria, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Pakistan.svg|thumb|Terrorism deaths in Pakistan, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Yemen.svg|thumb|Terrorism deaths in Yemen, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Somalia.svg|thumb|Terrorism deaths in Somalia, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Ukraine.svg|thumb|Terrorism deaths in Ukraine, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Sudan.svg|thumb|Terrorism deaths in Sudan and South Sudan, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Cameroon.svg|thumb|Terrorism deaths in Cameroon, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Libya.svg|thumb|Terrorism deaths in Libya, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] [[File:Terrorism deaths in Egypt.svg|thumb|Terrorism deaths in Egypt, 1970-2015 per the Global Terrorism Database.<ref name=Graves2017/>]] == Appendix 2. Democratization 1 and 10 years after the end of a conflict == The two figures in this appendix are similar to Figure 6 in the text. That shows the level of democracy 5 years after the end of a conflict vs. 1 year before. The two in this appendix show the level of democracy 1 and 10 years after the end of a conflict vs. 1 year before. [[File:Democratization 1 year after vs. 1 year before twentieth century revolutions.svg|thumb|Democratization 1 year after (vertical scale) vs. 1 year before (horizontal scale) twentieth century revolutions]] [[File:Democratization 10 years after vs. 1 year before twentieth century revolutions.svg|thumb|Democratization 10 years after (vertical scale) vs. 1 year before (horizontal scale) twentieth century revolutions]] == Notes == {{reflist}} [[Category:Original research]] [[Category:Research]] [[Category:Political science]] [[Category:Military]] [[Category:Military Science]] [[Category:Freedom and abundance]] jz70s0mklf86dbhanwrduvsi171ecqh 0 231176 2691720 2287917 2024-12-12T23:24:49Z Yinuo Liu 2935880 added instrument table 2691720 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{Wikipedia2|Swanson, Nolan and Pelham Teacher and Parent Rating Scale}} The '''Swanson, Nolan and Pelham Teacher and Parent Rating Scale (SNAP)''', developed by James Swanson, Edith Nolan and William Pelham, is a 90-question self-report inventory designed to measure attention deficit hyperactivity disorder (ADHD) and oppositional defiant disorder (ODD) symptoms in children and young adults. The questionnaire takes about 10 minutes to complete and is designed for use with children and young adults ages 6–18.<ref>{{cite journal|last1=Atkins|first1=MS|last2=Pelham|first2=WE|last3=Licht|first3=MH|title=A comparison of objective classroom measures and teacher ratings of Attention Deficit Disorder.|journal=Journal of abnormal child psychology|date=March 1985|volume=13|issue=1|pages=155–67|pmid=3973249|doi=10.1007/bf00918379}}</ref> == Scoring and interpretation == Scoring the SNAP-IV is based on a 0-3 scale, with each question being scored as follows based on participant response: * '''0 points:''' "not at all" * '''1 point:''' "just a little" * '''2 points:''' "quite a bit" * '''3 points:''' "very much" === Item breakdown === The questions measure different domains of ADHD and ODD. The breakdown is as follows: * '''1-10''': Measures attention deficit hyperactivity disorder inattention symptoms * '''11-20''': Measures attention deficit hyperactivity disorder hyperactivity/impulsivity symptoms * '''21-30''': Measures ODD symptoms * '''31-40''': Measures "general childhood problems" * '''41-80''': Measures non-ADHD disorders * '''81-90''': Measures academic performance and deportment === Interpretation of subscale scores === Subscale scores add all scores on the items in the subset and divided by the total number of items in the subset. Subscale score cutoffs for the disorders are as follows: * '''ADHD inattentive type''': Teacher score of 2.56, parent score of 1.78. * '''ADHD hyperactive/impulsive type''': Teacher score of 1.78, parent score of 1.44. * '''ADHD combined type''': Teacher score of 2.00, parent score of 1.67. * '''ODD''': Teacher score of 1.38, parent score of 1.88. == External Links == *[https://sccap53.org Society of Clinical Child and Adolescent Psychology] *[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/inattention-and-hyperactivity-adhd/ EffectiveChildTherapy.Org information on ADHD] == References == {{reflist}} {{:{{BASEPAGENAME}}/Navbox}} [[Category:Psychological measures]] tad9swz8bt52la2c43e1lqix9vo42b4 0 231179 2691719 2424135 2024-12-12T23:23:49Z Yinuo Liu 2935880 added instrument table 2691719 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{Wikipedia2|Strengths and Difficulties Questionnaire}} The Strengths and Difficulties Questionnaire (SDQ) is a behavioral screening questionnaire for children and adolescents ages 2 through 17 years old, developed by child psychiatrist [[w:Robert N. Goodman|Robert Goodman]] in the [[w:United Kingdom|United Kingdom]]. Versions of it are available for use for no fee. The combination of its brevity and noncommercial distribution have made it popular among clinicians and researchers. There are more than 3000 peer reviewed articles using it that are indexed in PubMed alone. Overall, the SDQ has been proven to have satisfactory construct and concurrent validity across a wide range of settings and samples.<ref>{{Cite journal|last=Stone|first=Lisanne L.|last2=Otten|first2=Roy|last3=Engels|first3=Rutger C. M. E.|last4=Vermulst|first4=Ad A.|last5=Janssens|first5=Jan M. A. M.|date=2010-09-01|title=Psychometric Properties of the Parent and Teacher Versions of the Strengths and Difficulties Questionnaire for 4- to 12-Year-Olds: A Review|journal=Clinical Child and Family Psychology Review|language=en|volume=13|issue=3|pages=254–274|doi=10.1007/s10567-010-0071-2|issn=1096-4037|pmc=2919684|pmid=20589428}}</ref><ref>{{Cite journal|last=Goodman|first=R.|last2=Meltzer|first2=H.|last3=Bailey|first3=V.|date=1998-10-01|title=The strengths and difficulties questionnaire: A pilot study on the validity of the self-report version|journal=European Child & Adolescent Psychiatry|language=en|volume=7|issue=3|pages=125–130|doi=10.1007/s007870050057|issn=1018-8827}}</ref><ref>{{Cite journal|last=Warnick|first=Erin M.|last2=Bracken|first2=Michael B.|last3=Kasl|first3=Stanislav|date=September 2008|title=Screening Efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: A Systematic Review|url=https://onlinelibrary.wiley.com/doi/10.1111/j.1475-3588.2007.00461.x|journal=Child and Adolescent Mental Health|language=en|volume=13|issue=3|pages=140–147|doi=10.1111/j.1475-3588.2007.00461.x}}</ref> It is considered a good general screening measure for attention problems,<ref>Owens, J. S., Evans, S. W., & Margherio, S. M. (2020). Assessment of attention deficit hyperactivity disorder. In E. A. Youngstrom, M. J. Prinstein, E. J. Mash, & R. Barkley (Eds.), Assessment of Disorders in Childhood and Adolescence (5th ed., pp. 93-131). Guilford Press.</ref> although the [[w:Sensitivity and specificity|sensitivity and specificity]] are not both over .80 at any single cut score, so it should not be used by itself as the basis for a diagnosis of [[w:Attention-deficit/hyperactivity disorder|attention-deficit/hyperactivity disorder]].<ref>{{Cite journal|last=Mulraney|first=Melissa|last2=Arrondo|first2=Gonzalo|last3=Musullulu|first3=Hande|last4=Iturmendi-Sabater|first4=Iciar|last5=Cortese|first5=Samuele|last6=Westwood|first6=Samuel J.|last7=Donno|first7=Federica|last8=Banaschewski|first8=Tobias|last9=Simonoff|first9=Emily|date=2022-08|title=Systematic Review and Meta-analysis: Screening Tools for Attention-Deficit/Hyperactivity Disorder in Children and Adolescents|url=https://linkinghub.elsevier.com/retrieve/pii/S0890856721020840|journal=Journal of the American Academy of Child & Adolescent Psychiatry|language=en|volume=61|issue=8|pages=982–996|doi=10.1016/j.jaac.2021.11.031}}</ref> There are versions of the SDQ designed for use in different situations, including a short form, a longer form with an impact supplement, and a follow-up form designed for use after a behavioral intervention. The questionnaire takes 3–10 minutes to complete. There are now self-report (completed by the youth), parent-report, and teacher-report versions. A version designed for adults (age 18+ years) to fill out about themselves has also been developed. The SDQ has been translated into more than 80 languages, including Spanish, Chinese, Russian, and Portuguese.<ref name="SDQ website">{{cite web|url=http://sdqinfo.org/py/sdqinfo/b0.py|title=Strengths and Difficulties Questionnaire|website=Strengths and Difficulties Questionnaire|accessdate=10 July 2015}}</ref> General population norms are available for the USA and UK for some of the variations of the SDQ. == Scoring and interpretation == Respondents are asked to indicate whether a specific attribute is “not true”, “somewhat true”, or “certainly true”. There are four versions of the short form: *Three parent/teacher versions for children ages 2–4, 4-10, and 11-17, *One self-report version for children ages 11–17. Scores on the attributes section range from 0-40, with [[prosocial behavior]] scale scores not included in the total score. All versions of the SDQ ask 25 questions sorted into 5 scales. Four of these are potential problems, and one is strength-related. {| class="wikitable" |- ! Subscale <br> (rated 0-2 for each question, maximum score of 10) !! colspan="2" span style="font-size:110%; text-align:center;" | Score and description !! colspan="1" span style="font-size:110%; text-align:center;" | Notes |- | '''Emotional symptoms subscale''' <br> (Questions 1-5)|| <u>'''Parent report''':</u> <br> '''0-3''': Normal behavior <br> '''4:''' Borderline abnormal behavior <br> '''5-10:''' Abnormal behavior || <u>'''Teacher report'''</u>: <br> '''0-4''': Normal behavior <br> '''5:''' Borderline abnormal behavior <br> '''6-10:''' Abnormal behavior || |- | '''Conduct problems subscale''' <br> (Questions 6-10) ||!! colspan="2" span style="font-size:100%; text-align:center;" | <u>'''Parent or Teacher report''':</u> <br> '''0-2''': Normal behavior <br> '''3:''' Borderline abnormal behavior <br> '''4-10:''' Abnormal behavior || Question 7 reverse scored* |- | '''Hyperactivity/inattention subscale''' <br> (Questions 11-15) || !! colspan="2" span style="font-size:100%; text-align:center;" | <u>'''Parent or Teacher report''':</u> <br> '''0-5''': Normal behavior <br> '''6:''' Borderline abnormal behavior <br> '''7-10:''' Abnormal behavior || Questions 14 and 15 reverse scored* |- | '''Peer relationship problems subscale''' <br> (Questions 16-20) || <u>'''Parent report''':</u> <br> '''0-2''': Normal behavior <br> '''3:''' Borderline abnormal behavior <br> '''4-10:''' Abnormal behavior || <u>'''Teacher report'''</u>: <br> '''0-3''': Normal behavior <br> '''4:''' Borderline abnormal behavior <br> '''5-10:''' Abnormal behavior || Questions 17 and 18 reverse scored* |- | '''Prosocial behavior subscale''' <br> (Questions 21-25) || !! colspan="2" span style="font-size:100%; text-align:center;" | <u>'''Parent or Teacher report''':</u> <br> '''6-10''': Normal behavior <br> '''5:''' Borderline abnormal behavior <br> '''0-4:''' Abnormal behavior || |- | '''Internalizing score''' <br> (summing emotional symptoms and peer relationship problems) || !! colspan="2" span style="font-size:100%; text-align:center;" | Not available || |- | '''Externalizing score''' <br> (summing conduct problems and hyperactivity/inattention) || !! colspan="2" span style="font-size:100%; text-align:center;" | Not available || |- | '''Total score''' <br> (summing scores from questions 1-20) || <u>'''Parent report''':</u> <br> '''0-13''': Normal behavior <br> '''14-16:''' Borderline abnormal behavior <br> '''17 and above:''' Abnormal behavior || <u>'''Teacher report'''</u>: <br> '''0-11''': Normal behavior <br> '''12-15:''' Borderline abnormal behavior <br> '''15 and above:''' Abnormal behavior || |} *<span style="font-size:88%"> *Reverse scoring means that responses of “not true” receive 2 points, responses of “somewhat true” receive 1 point, and responses of “certainly true” receive 0 points.</span> == Other scoring == The same 25 items are included in questionnaires for completion by the parents or teachers of 4- to 16-year-olds.<ref name="Goodman, R. 1997. p. 581-6">Goodman, R., The Strengths and Difficulties Questionnaire: a research note. J Child Psychol Psychiatry, 1997. 38(5): p. 581-6</ref> A slightly modified informant-rated version for the parents or nursery teachers of 2- to 4-year-olds. 22 items are identical, the item on reflectiveness is softened, and 2 items on antisocial behavior are replaced by items on oppositionality. Questionnaires for self-completion by adolescents ask about the same 25 traits, though the wording is slightly different as it is in the first person (e.g. 'I often worry' instead of 'Many worries, often seems worried'). This self-report version is suitable for young people aged around 11-17, depending on their level of understanding and literacy. ===B. An impact supplement=== Several two-sided versions of the SDQ are available with the 25 items on strengths and difficulties on the front of the page and an impact supplement on the back. These extended versions of the SDQ ask whether the respondent thinks the young person has a problem, and if so, enquire further about chronicity, distress, impairment in everyday activities, and burden to others. This provides useful additional information for clinicians and researchers with an interest in psychiatric caseness and the determinants of service use.<ref>Goodman, R., The extended version of the Strengths and Difficulties Questionnaire as a guide to child psychiatric caseness and consequent burden. J Child Psychol Psychiatry, 1999. 40(5): p. 791-9.</ref> ===C. Follow-up questions=== The follow-up versions of the SDQ include not only the 25 basic items and the impact question, but also two additional follow-up questions for use after an intervention of after visits to a clinic: # How have the intervention / clinic visits changed the child's problems? # Has the intervention helped in other ways, e.g. making the problems more bearable? To increase the chance of detecting change, the follow-up versions of the SDQ ask about 'the last month', as opposed to 'the last six months or this school year', which is the reference period for the standard versions. Follow-up versions also omit the question about the chronicity of problems. ==How to score the SDQ== The SDQ scoring site allows one to score paper copies of a parent, teacher and/or self-report SDQ free-of-charge, and generates a brief report. Briefly, each of the five scales of the SDQ are scored from 0-10, and one can add up four of these (emotional, conduct, hyperactivity and peer problems) to create a '''total difficulty score''' (range 0-40).<ref name="Goodman, R. 1997. p. 581-6"/> One can also add the emotional and peer items together to get an internalizing problems score (range 0-20) and add the conduct and hyperactivity questions together to get an externalizing score (range 0-20). [[Category:Psychological measures]] == External links == * {{Official website|http://www.sdqinfo.org/}} * [https://pubmed.ncbi.nlm.nih.gov/?term=%22Strengths+and+Difficulties%22+or+SDQ&sort= Articles using the SDQ indexed in PubMed] * [http://effectivechildtherapy.org/concerns-symptoms-disorders/ EffectiveChildTherapy.Org information on concerns, symptoms, and disorders] * [https://sccap53.org Society of Clinical Child and Adolescent Psychology] [[Category:Screening and assessment tools in child and adolescent psychiatry]] ==References== {{reflist}} {{:{{BASEPAGENAME}}/Navbox}} rld4qzwwuojaun837gh9hhsanesgh4u 0 232469 2691716 2691537 2024-12-12T23:16:51Z Young1lim 21186 /* ARM Assembly Programming (II) */ 2691716 wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20230726.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Branch and Return Methods ([[Media:ARM.2ASM.Branch.20241213.pdf |pdf]]) * 2. PC Relative Addressing ([[Media:ARM.2ASM.PCRelative.20241123.pdf |pdf]]) * 3. Thumb instruction Set ([[Media:ARM.2ASM.Thumb.20241123.pdf |pdf]]) * 4. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 5. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 6. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 7. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 8. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 9. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20230610.pdf |pdf]]) * 10. Tail Chaining ([[Media:ARM.2ASM.TailChain.20230816.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220924.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] c1ch0ekbb7d92mseb4zma4buqobdegi 1 243667 2691640 1962254 2024-12-12T15:20:42Z MMCLXXII 2994915 /* propaganda and disinformation looms maybe */ new section 2691640 wikitext text/x-wiki == Knowing how you know == As fake news becomes increasingly entangled with real news, each of us has a responsibility to [[Knowing How You Know|know how we know]]. This is difficult, but could become easier if social media sites required [[w:Traceability|traceability]] to the original author for every post. The idea is: 1) Each social media site requires verification that each user is an identifiable real person. 2) The [[w:Author|author]] of each post is identified when the post originates. 3) A log entry is created as each original post is re-posted. This allows every post to be traced back to the original author, who has been verified to be a real person. 4) When a post originates from an organization, rather than an individual, the organization is identified as journalism or not. If an organization claims to be journalism, then their [[w:Journalism_ethics_and_standards|journalism policy]] must be published transparently. This could help us identify reliable sources from unreliable sources. When the framers of the Bill of Rights protected freedom of speech, the author of each speech was easily determined. This proposal could help restore that accountability in our age of social media. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:16, 2 January 2019 (UTC) :Have you seen {{cite Q|Q56027099}}? This book claims that Facebook has so much data on so many people, that it becomes profitable to target ads to groups as small as 20. Each such ad could seem complete ridiculous to 99 percent of humanity but resonate with the preconceptions of those 20. In so doing, it could push them to become even more extreme. This combined with the research of {{w|Daniel Kahneman}} and summarized in his (2011) ''{{w|Thinking, Fast and Slow}}'' explains how Facebook, and to a lesser extent other media, make money by amplifying the Balkanization and exploitation of the international body politic. :I think this calls for a couple of things: :# Require all media companies to maintain a database of all ads open to the public with the text and the sources of the ads fully searchable and otherwise fully available for data mining by others -- and requiring all advertisers to identify the real sources of the money, not creating millions of shell companies for different ads. :# A social movement to encourage people to migrate away from for-profit social media to free open-source social media, based on software like {{w|Mastodon (software)}}, {{w|Diaspora (software)}}, {{w|GNU social}}, and others on the Wikipedia [[w:Comparison of software and protocols for distributed social networking]] and [[w:Comparison of microblogging services]]. I have not personally tried any of these, but I think it could be quite feasible to convince a critical mass of the international body politic to move away from for-profit social media to non-profit social media, as described in this article. :Thanks for your comments. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 16:33, 2 January 2019 (UTC) :: Mark Twain and others observed that "A lie can travel halfway around the world before the truth can get its boots on." Therefore it seems more important to reduce the allure of falsehoods than to increase access to accurate (and representative) information. Therefore I continue to focus on undercutting disinformation by inoculating the consumer against nonsense. My courses on [[Knowing How You Know|Knowing How you Know]], [[Intellectual Honesty]], [[Practicing Dialogue]], [[Socratic Methods]] and the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking]] curriculum all seek to do that. Enrollment is underwhelming. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:39, 3 January 2019 (UTC) == propaganda and disinformation looms maybe == basically this: if we're here to publish news that matter to us and some of them make the cut while most of them don't (according to the page stats from 2017, 3/4ths won't see the light of the day globally) what prevents a group of trolls/attackers/etc to log in as a unit with multiple accounts and publish whatever they want? and similarly what prevents the editors (i mean the people who decide what gets published, sorry i'm new and not very terminologically informed) from being impartial? who gets to decide what views are represented here and on what bases? idk. maybe i'm asking this for the wrong wiki, maybe this is a topic for wikinews but i surely can't tell the difference if there is one. i need answers, and please don't tell me "uhh there's a code of conduct" because we all know that the line gets *very* blurry for so-called "controversial" topics. idk the definitions do not seem rigidly drawn and that's a trust issue for me. [[User:MMCLXXII|MMCLXXII]] ([[User talk:MMCLXXII|discuss]] • [[Special:Contributions/MMCLXXII|contribs]]) 15:20, 12 December 2024 (UTC) 88gc7eta0bgbcpj635a7494qf2w4okq 2691647 2691640 2024-12-12T16:17:46Z DavidMCEddy 218607 /* propaganda and disinformation looms maybe */ 2691647 wikitext text/x-wiki == Knowing how you know == As fake news becomes increasingly entangled with real news, each of us has a responsibility to [[Knowing How You Know|know how we know]]. This is difficult, but could become easier if social media sites required [[w:Traceability|traceability]] to the original author for every post. The idea is: 1) Each social media site requires verification that each user is an identifiable real person. 2) The [[w:Author|author]] of each post is identified when the post originates. 3) A log entry is created as each original post is re-posted. This allows every post to be traced back to the original author, who has been verified to be a real person. 4) When a post originates from an organization, rather than an individual, the organization is identified as journalism or not. If an organization claims to be journalism, then their [[w:Journalism_ethics_and_standards|journalism policy]] must be published transparently. This could help us identify reliable sources from unreliable sources. When the framers of the Bill of Rights protected freedom of speech, the author of each speech was easily determined. This proposal could help restore that accountability in our age of social media. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:16, 2 January 2019 (UTC) :Have you seen {{cite Q|Q56027099}}? This book claims that Facebook has so much data on so many people, that it becomes profitable to target ads to groups as small as 20. Each such ad could seem complete ridiculous to 99 percent of humanity but resonate with the preconceptions of those 20. In so doing, it could push them to become even more extreme. This combined with the research of {{w|Daniel Kahneman}} and summarized in his (2011) ''{{w|Thinking, Fast and Slow}}'' explains how Facebook, and to a lesser extent other media, make money by amplifying the Balkanization and exploitation of the international body politic. :I think this calls for a couple of things: :# Require all media companies to maintain a database of all ads open to the public with the text and the sources of the ads fully searchable and otherwise fully available for data mining by others -- and requiring all advertisers to identify the real sources of the money, not creating millions of shell companies for different ads. :# A social movement to encourage people to migrate away from for-profit social media to free open-source social media, based on software like {{w|Mastodon (software)}}, {{w|Diaspora (software)}}, {{w|GNU social}}, and others on the Wikipedia [[w:Comparison of software and protocols for distributed social networking]] and [[w:Comparison of microblogging services]]. I have not personally tried any of these, but I think it could be quite feasible to convince a critical mass of the international body politic to move away from for-profit social media to non-profit social media, as described in this article. :Thanks for your comments. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 16:33, 2 January 2019 (UTC) :: Mark Twain and others observed that "A lie can travel halfway around the world before the truth can get its boots on." Therefore it seems more important to reduce the allure of falsehoods than to increase access to accurate (and representative) information. Therefore I continue to focus on undercutting disinformation by inoculating the consumer against nonsense. My courses on [[Knowing How You Know|Knowing How you Know]], [[Intellectual Honesty]], [[Practicing Dialogue]], [[Socratic Methods]] and the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking]] curriculum all seek to do that. Enrollment is underwhelming. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:39, 3 January 2019 (UTC) == propaganda and disinformation looms maybe == basically this: if we're here to publish news that matter to us and some of them make the cut while most of them don't (according to the page stats from 2017, 3/4ths won't see the light of the day globally) what prevents a group of trolls/attackers/etc to log in as a unit with multiple accounts and publish whatever they want? and similarly what prevents the editors (i mean the people who decide what gets published, sorry i'm new and not very terminologically informed) from being impartial? who gets to decide what views are represented here and on what bases? idk. maybe i'm asking this for the wrong wiki, maybe this is a topic for wikinews but i surely can't tell the difference if there is one. i need answers, and please don't tell me "uhh there's a code of conduct" because we all know that the line gets *very* blurry for so-called "controversial" topics. idk the definitions do not seem rigidly drawn and that's a trust issue for me. [[User:MMCLXXII|MMCLXXII]] ([[User talk:MMCLXXII|discuss]] • [[Special:Contributions/MMCLXXII|contribs]]) 15:20, 12 December 2024 (UTC) :{{re|MMCLXXII}} You have identified a real issue. Wikimedia Foundation projects do reasonably well in this regard with their rules that allow almost anyone to change almost anything while writing from a neutral point of vie citing credible sources. It's not perfect: Ideologues and paid trolls sometimes win. The Wikipedia article on [[w: Wikipedia and the Israeli–Palestinian conflict| Wikipedia and the Israeli–Palestinian conflict]] discusses some of the problems and how the those problems have been managed so far. :On the whole, I would say that Wikimedia Foundation projects are far from perfect but better than the alternatives, because the honest volunteer editors have so far been winning in most cases. We need more honest volunteer editors, because by many powerful people recognize Wikipedia as a threat and will pay more trolls and recruit more ideologues as they increasing figure out how to do that and share that information with other powerful people. :My summary of <!--Ressa (2022) How to Stand Up To a Dictator-->{{cite Q|Q117559286}} is as follows: :{{quote| * Filipino teenagers were world leaders in figuring out how to make Facebook and Twitter boost whatever message they wanted. (pp. 125-6) * Those methods are being used worldwide to build support for fascism and destroy democracy. * Duterte’s social media campaign ... started ... with five hundred volunteers (pp. 147-8) * (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact the targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest. (pp. 152-3) }} :With these methods, Philippine President [[w:Rodrigo Duterte|Rodrigo Duterte]] was able to get [[w:Maria Ressa|Maria Ressa]] prosecuted and convicted of a felony on trumped up charges. She won the 2021 Nobel Peace Prize (with a Russian journalist) for her resistance. The last I heard, she is still fighting that conviction while flying all of the world working to make it harder for ideologues and fascists to win. :On May 23, 2013, then-US President Obama noted that terrorism caused fewer American deaths than car accidents or falls in the bathtub. He occasionally ''had to be badgered by advisors into choices commensurate with popular fear'', as noted in "[[Winning the War on Terror]]". Yet we do NOT declare war on bathtubs. :See also [[Information is a public good: Designing experiments to improve government]]. :Thanks for your comment. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 16:17, 12 December 2024 (UTC) e7g1dq6k80ijd614yr8n1mu248j0pi5 2691650 2691647 2024-12-12T16:27:06Z DavidMCEddy 218607 /* propaganda and disinformation looms maybe */ typo 2691650 wikitext text/x-wiki == Knowing how you know == As fake news becomes increasingly entangled with real news, each of us has a responsibility to [[Knowing How You Know|know how we know]]. This is difficult, but could become easier if social media sites required [[w:Traceability|traceability]] to the original author for every post. The idea is: 1) Each social media site requires verification that each user is an identifiable real person. 2) The [[w:Author|author]] of each post is identified when the post originates. 3) A log entry is created as each original post is re-posted. This allows every post to be traced back to the original author, who has been verified to be a real person. 4) When a post originates from an organization, rather than an individual, the organization is identified as journalism or not. If an organization claims to be journalism, then their [[w:Journalism_ethics_and_standards|journalism policy]] must be published transparently. This could help us identify reliable sources from unreliable sources. When the framers of the Bill of Rights protected freedom of speech, the author of each speech was easily determined. This proposal could help restore that accountability in our age of social media. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:16, 2 January 2019 (UTC) :Have you seen {{cite Q|Q56027099}}? This book claims that Facebook has so much data on so many people, that it becomes profitable to target ads to groups as small as 20. Each such ad could seem complete ridiculous to 99 percent of humanity but resonate with the preconceptions of those 20. In so doing, it could push them to become even more extreme. This combined with the research of {{w|Daniel Kahneman}} and summarized in his (2011) ''{{w|Thinking, Fast and Slow}}'' explains how Facebook, and to a lesser extent other media, make money by amplifying the Balkanization and exploitation of the international body politic. :I think this calls for a couple of things: :# Require all media companies to maintain a database of all ads open to the public with the text and the sources of the ads fully searchable and otherwise fully available for data mining by others -- and requiring all advertisers to identify the real sources of the money, not creating millions of shell companies for different ads. :# A social movement to encourage people to migrate away from for-profit social media to free open-source social media, based on software like {{w|Mastodon (software)}}, {{w|Diaspora (software)}}, {{w|GNU social}}, and others on the Wikipedia [[w:Comparison of software and protocols for distributed social networking]] and [[w:Comparison of microblogging services]]. I have not personally tried any of these, but I think it could be quite feasible to convince a critical mass of the international body politic to move away from for-profit social media to non-profit social media, as described in this article. :Thanks for your comments. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 16:33, 2 January 2019 (UTC) :: Mark Twain and others observed that "A lie can travel halfway around the world before the truth can get its boots on." Therefore it seems more important to reduce the allure of falsehoods than to increase access to accurate (and representative) information. Therefore I continue to focus on undercutting disinformation by inoculating the consumer against nonsense. My courses on [[Knowing How You Know|Knowing How you Know]], [[Intellectual Honesty]], [[Practicing Dialogue]], [[Socratic Methods]] and the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking]] curriculum all seek to do that. Enrollment is underwhelming. Thanks! --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:39, 3 January 2019 (UTC) == propaganda and disinformation looms maybe == basically this: if we're here to publish news that matter to us and some of them make the cut while most of them don't (according to the page stats from 2017, 3/4ths won't see the light of the day globally) what prevents a group of trolls/attackers/etc to log in as a unit with multiple accounts and publish whatever they want? and similarly what prevents the editors (i mean the people who decide what gets published, sorry i'm new and not very terminologically informed) from being impartial? who gets to decide what views are represented here and on what bases? idk. maybe i'm asking this for the wrong wiki, maybe this is a topic for wikinews but i surely can't tell the difference if there is one. i need answers, and please don't tell me "uhh there's a code of conduct" because we all know that the line gets *very* blurry for so-called "controversial" topics. idk the definitions do not seem rigidly drawn and that's a trust issue for me. [[User:MMCLXXII|MMCLXXII]] ([[User talk:MMCLXXII|discuss]] • [[Special:Contributions/MMCLXXII|contribs]]) 15:20, 12 December 2024 (UTC) :{{re|MMCLXXII}} You have identified a real issue. Wikimedia Foundation projects do reasonably well in this regard with their rules that allow almost anyone to change almost anything while writing from a neutral point of vie citing credible sources. It's not perfect: Ideologues and paid trolls sometimes win. The Wikipedia article on [[w: Wikipedia and the Israeli–Palestinian conflict| Wikipedia and the Israeli–Palestinian conflict]] discusses some of the problems and how the those problems have been managed so far. :On the whole, I would say that Wikimedia Foundation projects are far from perfect but better than the alternatives, because the honest volunteer editors have so far been winning in most cases. We need more honest volunteer editors, because many powerful people recognize Wikipedia as a threat and will pay more trolls and recruit more ideologues as they increasing figure out how to do that and share that information with other powerful people. :My summary of <!--Ressa (2022) How to Stand Up To a Dictator-->{{cite Q|Q117559286}} is as follows: :{{quote| * Filipino teenagers were world leaders in figuring out how to make Facebook and Twitter boost whatever message they wanted. (pp. 125-6) * Those methods are being used worldwide to build support for fascism and destroy democracy. * Duterte’s social media campaign ... started ... with five hundred volunteers (pp. 147-8) * (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact the targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest. (pp. 152-3) }} :With these methods, Philippine President [[w:Rodrigo Duterte|Rodrigo Duterte]] was able to get [[w:Maria Ressa|Maria Ressa]] prosecuted and convicted of a felony on trumped up charges. She won the 2021 Nobel Peace Prize (with a Russian journalist) for her resistance. The last I heard, she is still fighting that conviction while flying all of the world working to make it harder for ideologues and fascists to win. :On May 23, 2013, then-US President Obama noted that terrorism caused fewer American deaths than car accidents or falls in the bathtub. He occasionally ''had to be badgered by advisors into choices commensurate with popular fear'', as noted in "[[Winning the War on Terror]]". Yet we do NOT declare war on bathtubs. :See also [[Information is a public good: Designing experiments to improve government]]. :Thanks for your comment. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 16:17, 12 December 2024 (UTC) mnmnjkn344y129k0ps33ng7q4bkcob4 2 246046 2691688 2629614 2024-12-12T21:36:50Z Dc.samizdat 2856930 /* Research articles */ 2691688 wikitext text/x-wiki I am David Brooks Christie, born April 3, 1951. <blockquote>He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.<ref>{{Cite book|title=By His Bootstraps|last=Heinlein|first=Robert|work=The Menace From Earth|publisher=|year=1959|isbn=|location=|pages=|author-link=W:Robert Heinlein}}</ref> </blockquote> and yet... <blockquote>Oh, what good it does the heart, to know it isn't magic!<ref>{{Cite book|url=https://books.google.com/books?id=R1HTBQAAQBAJ&pg=PT43&lpg=PT43&dq=I+used+to+imagine+him+coming+from+his+house,+like+Merlin&source=bl&ots=h707RlOyPN&sig=ACfU3U22OoWoBsZwY2B-Si_jc1jdy-re1w&hl=en&sa=X&ved=2ahUKEwj06-ebwZTiAhVFvZ4KHXVkAaMQ6AEwAHoECAcQAQ#v=onepage&q=I%20used%20to%20imagine%20him%20coming%20from%20his%20house%2C%20like%20Merlin&f=false|title=Stanley Kunitz|last=Oliver|first=Mary|work=Dream Work|publisher=|year=1986|isbn=|location=|pages=|author-link=W:Mary Oliver}}</ref> </blockquote> == Polyscheme project == {{Polyscheme}} The [[Polyscheme]] project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes that pop-up in context. Some of what's in these companion articles is opinion, not established fact, as of this date of publication. == Research articles == Wiki articles I write or contribute to which contain original research or commentary, and so cannot be published as [[wikipedia:No original research|Wikipedia articles]], are hosted here at Wikiversity instead. When complete they can be submitted for peer review, and in some cases for publication elsewhere. Perhaps eventually some will even qualify to be cited as sources by Wikipedia articles. [[WikiJournal Preprints/Kinematics of the cuboctahedron]] [[WikiJournal Preprints/24-cell]] == History == I was influenced by the works of [[W:Buckminster Fuller|Buckminster Fuller]], whose ''Education Automation''<ref>{{Cite book|title=Education Automation: Comprehensive Learning for Emergent Humanity|last=Fuller|first=R. Buckminster|author-link=W:Buckminster Fuller|year=1962|editor-last=Snyder|editor-first=Jaime|publisher=Lars Müller|url=https://www.lars-mueller-publishers.com/education-automation}}</ref> I discovered in college at [[W:Oberlin college|Oberlin]]. Over the next years I read everything he published, and built a [[W:Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]] tree house centered 73.5 feet up between the twin trunks of a white pine tree in Vermont. I once walked in at the middle of one of Bucky's famous 5-hour lectures about everything, at U.C. Santa Barbara on my first trip to California in January 1971. Later I taught an Experimental College course at Oberlin on Fuller's works, where we built and flew a geodesic hot air balloon that we watched fly away to Canada over Lake Erie. We didn't think it set anything in Canada on fire with its open flame burner, because it didn't appear to start its descent until it had exhausted its fuel. After college I moved to Chicago, where I did programming in assembly language for one of the first large (nation-wide) online database transaction processing systems, hosted by a single IBM System/370 mainframe that occupied an entire floor of an office tower in the downtown Chicago loop. From Chicago I led summer-long Wilderness Projects in canoes through the Canadian taiga, straddling the 60th parallel near where the Northwest Territories, Nunavut, Manitoba and Saskatchewan meet clockwise at a four-corners, 1400 miles northwest of Chicago, 400 miles north of the northernmost town you can reach by road, and 500 miles north of our nearest radio contact, in those days before satellite phones. In 1978 I landed a programming job in California working at a tiny microcomputer manufacturer, one of the earliest such start-ups, the year after the Apple II personal computer was introduced, three years before the advent of the IBM PC. In 1982 I reached Silicon Valley, the end of the rainbow for ambitious programmers, epicenter of the emergence, swiftly urbanizing frontier homestead of the digital revolution. We lived in a refurbished cabin in a redwood forest on the wild San Mateo County coast, and I commuted over-the-hill every day to my job at a computer manufacturer, in the same truck I had commuted to Canada in with 3 canoes on the roof every summer from Chicago. At [[W:Convergent Technologies|that company]], the job I held longest in my peripatetic career (7 years) and learned the most from, they built the first fully concurrent, ethernet-networked office workstations from Intel's newest microprocessors (before the silicon was even dry and fully debugged), and I wrote code for the internals of their distributed, message-based operating system. At last I had arrived at a long-sought destination, the root systems of the computer. In Silicon Valley I worked for stock options at a series of interesting start-ups that did not work out, until one of them that was starting to work out was bought by [[W:Netscape|Netscape]] (for its engineering talent, not its software product) in the year of the Netscape IPO, 1995. This was a pretty good place to wind up, at the hottest start-up I had ever been early to, growing so fast it made our heads spin just to come to work the next day, at perhaps the most interesting nexus in Silicon Valley history thus far, the birth of the internet. But Netscape did not work out either in the end, as everybody knows. So I retired at the end of 1998 from writing programs other people wanted, to look for work that interested me. My principal interest had become studying symmetries in and among languages of non-deterministic computational mathematics. I had hard-won experience with computer programming languages and operating systems' internals, and a deep fascination with the problem of concurrency, but I had no mathematics. I had always disliked doing math homework assignments, the same way I detest repetitive unoriginal programming tasks.{{Efn|They are what computers are for.}} They were both hard for me for some reason.{{Efn|I think I may be algebraically dyslexic. If I don't have a picture of them, even double negatives make me work up a sweat.}} I could barely bring myself to do the grunt work my trade often demanded, and got into trouble at several companies by spending too much time making the assigned task interesting to me by biggering the design. After I failed the AP calculus exam in high school by failing to practice doing enough calculus, my formal education in mathematics ceased. At Netscape, where I was not a researcher just a programmer, and not of the original Netscape browser (Mosaic/Navigator/Mozilla), I had met the researcher [[W:Ramanathan V. Guha|R.V. Guha]], whose [[W:Meta Content Framework|Meta Content Framework (MCF)]] was in the process of being standardized as [[W:Resource Description Framework|Resource Description Framework (RDF)]], the semantic data representation that would later become a core component of Tim Berners-Lee's semantic web initiative. RDF is a language root system of semantic triples, subject-predicate-object, essentially the same data model used by the original AI researchers, who sought to construct language models by design, long before modern large language models were invented. Those are grown, not designed, from machine experience of the symmetries to be found in vast quantities of human speech found in nature. Modern AIs are not built by hand as RDF models by human architects; they are products of Darwinian natural selection, like us, rather than products of design, like the things we build. As a knowledge root system RDF interested me, but it has no operations, only a data model. After leaving Netscape I worked independently{{Efn|"Independently" in every sense: nobody paid me, I had no institutional affiliation of any kind any more, I had no colleagues any more, I published nothing and never even released any open-source code. I was independent in the sense of being completely solitary and invisible and unknowable, such that nobody could possibly have taken notice of my work even if they had been so inclined. I worked like Emily Dickinson during her lifetime, though with much less genius and prolificacy. But I was free and having fun and didn't care.}} for years programming a kind of symmetry group, consisting of the 3-dimensional RDF triples extended in a temporal 4th dimension like the one in Minkowski spacetime (the 4th field I added to the triple was a creation timestamp), and with [[W:David Gelernter|Gelernter]]'s four concurrency group operators rd(), out(), in(), and eval().{{Sfn|Carriero|Gelernter|1989|loc=The C-Linda programming language}} I realized my system as C++ template metaprograms{{Sfn|Stroustrup|2013|loc=''The [[W:C++|C++ Programming Language]]''}} for a nesting set of these operators, implemented as ACID-transactional C++ sequence iterators over the progressively more complex spaces traversed by each of Gelernter's operations. To this hierarchy I added an anonymous 5th operator()() (operator function call) between in() and eval(). None of this graph database transaction processing monitor{{Sfn|Gray|Reuter|1993|loc=''[[W:Transaction processing|Transaction processing]]''}} that I built bore any conscious resemblance to the sequence of 4-polytopes (hyper-polyhedra) of increasing complexity, with their 4th operator the 24-cell (hyper-anonymous), as I had not yet resumed my college-days' study of geometry. I had studied some physics (relativity) in college and understood it as the geometry of 4-dimensional space, I had wondered about Bucky Fuller's jitterbug ever since those days, and I had been pleased to "discover" Pascal's triangle of the ''n''-simplexes back in Chicago. But I had not read much of [[W:Regular Polytopes|Regular Polytopes]] yet, and had not made the acquaintance of the more astonishing objects [[W:Ludwig Schläfli|Schläfli]] discovered in 4-dimensional Euclidean space. Particularly not yet the unique 24-cell (hyper-cuboctahedron), the radially equilateral vector equilibrium Bucky Fuller saw the cuboctahedral shadow of. Fuller searched all his life for this object (the utterly unique realization of the 24-point symmetry group of the tetrahedron), but never quite found it because he was looking for it in the wrong space (3-space). .... == Nature is symmetry == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[wikipedia:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008|loc=''The Symmetries of Things''}} As I understand [[wikipedia:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[wikipedia:Theory of relativity|Einstein's relativity]] or [[wikipedia:Evolution|Darwin's evolution]] or [[wikipedia:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of the distinct [[wikipedia:symmetry group|symmetry groups]]. .... == Poetry is symmetry == [[W:Edna St. Vincent Millay|Edna St. Vincent Millay]] and many other [[W:Lawrence Ferlinghetti#Poetry|American and non-American poets]] knew that poetry is the Insurgent Art of inventing symmetries, and at its best contains a discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, [[W:Alicia Boole Stott|Alicia Boole Stott]]. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in [[W:George Dillon (poet)|George Dillon]] the soul of an earlier Parisian youth who burned brightly, [[W:Évariste Galois|Évariste Galois]] who discovered the mathematics which underlies geometry, inventing symmetry group theory before his own fatal interview at 20. Millay's contemporary poet [[W:Emmy Noether|Emmy Noether]], the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself an expression of an exact symmetry group. These poets knew how great poetry emerges from discovery, or rediscovery, of nature's symmetries. == Justice is symmetry == Anyone should understand Israelis' unquenchable thirst for vengence for all acts that have attempted to exterminate them. Precisely because it is unquenchable, a survival instinct acquired at immeasurable cost, we must implacably resist, by all nonviolent means available to us, their attempt to slake it. Vengeance is mine, saith the Lord, and an eye for an eye makes the whole world blind. <blockquote> We are deceived into believing that we can get the kind of world we seek by doing the very things we are trying to get rid of. "Just a little more violence to end violence." "Just a little more hatred to end hatred." "Just a little more oppression to end oppression" -- and on and on. We are taken in because good people are doing these things, sincere and brave people. And this is why the finer their qualities, the more dangerous they are, the more thoroughly we are fooled. All the finest qualities in the world cannot change the simple, immutable fact that the ends cannot justify the means, but, on the contrary, the means determine the ends. In all of man's history this stands out clearly and intellectually indisputable; yet it has been perversely, insistently, sentimentally and tragically ignored. In this universe the means always and everywhere, without doubt and without exception, cannot, in the very nature of things, but determine the ends. This cannot be repeated often enough. We get what we do; not what we intend, dream, or desire. We simply get what we do. Recognizing this and applying it would, in a generation, bring about the transformation that alone can put an end to the fear, suspicion and misery which at present hold such terrible sway over all of our lives. If we see and act upon this (I will say again, unabashedly, what it is -- the means determine the ends!), then what the prophets of the ages have wistfully called Utopia will become a reality. “Nation shall no more lift sword against nation” nor unloose napalm, nerve gas or nuclear weapons. “Neither shall they learn war any more. But they shall sit every man under his vine and under his fig tree; and none shall make them afraid.” Because they will have at last understood, because we will all have at least understood, what is required of us. “To do justly and to love mercy, and to walk humbly” with the knowledge that all our means are but temporary ends and that all our ends are but new beginnings. We will have learned what every flower has never forgotten and what all oceans patiently remind us of. : - [[W:Ira Sandperl|Ira Sandperl]]{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14}} </blockquote> == Religion isn't symmetry == They say a dog is a man's best friend, but not every man should have a dog. It depends on the man. And the dog. If you want to have a dog, or a religion, as a companion and soulmate to help you answer important questions outside science's purvue, like how to be happy without making other people unhappy, have at it, and dog bless you. People who treat their religion as a source of facts about the world, instead of as a source of mystery, haven't received word yet that we have already passed through that revolutionary period in human history a few centuries back called [[w:Age_of_Enlightenment|the Enlightenment]]. Hello, we've discovered that the origin story of the facts is not mythologies, it's science. Just as you musn't let a dog drive your car or let a religion drive your government, you mustn't let a dog advise you on investment decisions or let a dogma dispute the facts that science has discovered. Religion has to stay in its lane. People who drive their religion weaving all over the road are a menace. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=[[W:Kepler's Books|Kepler's Books]]|place=Menlo Park, CA|isbn=0-8314-0035-8|jstor=73-93870|url=https://www.irasandperl.org/wordpress/index.php}} * {{Cite journal|last2=Gelernter|first2=David|author2-link=W:David Gelernter|last1=Carriero|first1=Nicholas|title=How to Write Parallel Programs: A Guide to the Perplexed|date=1989|journal=ACM Computing Surveys|volume=21|issue=3|url=https://dl.acm.org/doi/pdf/10.1145/72551.72553}} * {{Cite book|last=Stroustrup|first=Bjarne|title=The [[W:C++|C++ Programming Language]]: C++11|edition=4th|date=2013|author-link=W:Bjarne Stroustrup|publisher=Addison-Wesley}} * {{Cite book|last1=Gray|first1=Jim|author-link=W:Jim Gray (computer scientist)|last2=Reuter|first2=Andreas|title=Transaction Processing: Concepts and Techniques|title-link=W:Transaction processing|date=1993|publisher}=Morgan Kaufmann|place=San Mateo, CA}} {{Refend}} e40m0pmlbpowajxsr4uao495wqfmbwv 2691689 2691688 2024-12-12T21:44:36Z Dc.samizdat 2856930 /* History */ 2691689 wikitext text/x-wiki I am David Brooks Christie, born April 3, 1951. <blockquote>He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.<ref>{{Cite book|title=By His Bootstraps|last=Heinlein|first=Robert|work=The Menace From Earth|publisher=|year=1959|isbn=|location=|pages=|author-link=W:Robert Heinlein}}</ref> </blockquote> and yet... <blockquote>Oh, what good it does the heart, to know it isn't magic!<ref>{{Cite book|url=https://books.google.com/books?id=R1HTBQAAQBAJ&pg=PT43&lpg=PT43&dq=I+used+to+imagine+him+coming+from+his+house,+like+Merlin&source=bl&ots=h707RlOyPN&sig=ACfU3U22OoWoBsZwY2B-Si_jc1jdy-re1w&hl=en&sa=X&ved=2ahUKEwj06-ebwZTiAhVFvZ4KHXVkAaMQ6AEwAHoECAcQAQ#v=onepage&q=I%20used%20to%20imagine%20him%20coming%20from%20his%20house%2C%20like%20Merlin&f=false|title=Stanley Kunitz|last=Oliver|first=Mary|work=Dream Work|publisher=|year=1986|isbn=|location=|pages=|author-link=W:Mary Oliver}}</ref> </blockquote> == Polyscheme project == {{Polyscheme}} The [[Polyscheme]] project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes that pop-up in context. Some of what's in these companion articles is opinion, not established fact, as of this date of publication. == Research articles == Wiki articles I write or contribute to which contain original research or commentary, and so cannot be published as [[wikipedia:No original research|Wikipedia articles]], are hosted here at Wikiversity instead. When complete they can be submitted for peer review, and in some cases for publication elsewhere. Perhaps eventually some will even qualify to be cited as sources by Wikipedia articles. [[WikiJournal Preprints/Kinematics of the cuboctahedron]] [[WikiJournal Preprints/24-cell]] == History == I was influenced by the works of [[W:Buckminster Fuller|Buckminster Fuller]], whose ''Education Automation''<ref>{{Cite book|title=Education Automation: Comprehensive Learning for Emergent Humanity|last=Fuller|first=R. Buckminster|author-link=W:Buckminster Fuller|year=1962|editor-last=Snyder|editor-first=Jaime|publisher=Lars Müller|url=https://www.lars-mueller-publishers.com/education-automation}}</ref> I discovered in college at [[W:Oberlin college|Oberlin]]. Over the next years I read everything he published, and built a [[W:Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]] tree house centered 73.5 feet up between the twin trunks of a white pine tree in Vermont. I once walked in at the middle of one of Bucky's famous 5-hour lectures about everything, at U.C. Santa Barbara on my first trip to California in January 1971. Later I taught an Experimental College course at Oberlin on Fuller's works, where we built and flew a geodesic hot air balloon that we watched fly away to Canada over Lake Erie. We didn't think it set anything in Canada on fire with its open flame burner, because it didn't appear to start its descent until it had exhausted its fuel. After college I moved to Chicago, where I did programming in assembly language for one of the first large (nation-wide) online database transaction processing systems, hosted by a single IBM System/370 mainframe that occupied an entire floor of an office tower in the downtown Chicago loop. From Chicago I led summer-long Wilderness Projects in canoes through the Canadian taiga, straddling the 60th parallel near where the Northwest Territories, Nunavut, Manitoba and Saskatchewan meet clockwise at a four-corners, 1400 miles northwest of Chicago, 400 miles north of the northernmost town you can reach by road, and 500 miles north of our nearest radio contact, in those days before satellite phones. In 1978 I landed a programming job in California working at a tiny microcomputer manufacturer, one of the earliest such start-ups, the year after the Apple II personal computer was introduced, three years before the advent of the IBM PC. In 1982 I reached Silicon Valley, the end of the rainbow for ambitious programmers, epicenter of the emergence, swiftly urbanizing frontier homestead of the digital revolution. We lived in a refurbished cabin in a redwood forest on the wild San Mateo County coast, and I commuted over-the-hill every day to my job at a computer manufacturer, in the same truck I had commuted to Canada in with 3 canoes on the roof every summer from Chicago. At [[W:Convergent Technologies|that company]], the job I held longest in my peripatetic career (7 years) and learned the most from, they built the first fully concurrent, ethernet-networked office workstations from Intel's newest microprocessors (before the silicon was even dry and fully debugged), and I wrote code for the internals of their distributed, message-based operating system. At last I had arrived at a long-sought destination, the root systems of the computer. In Silicon Valley I worked for stock options at a series of interesting start-ups that did not work out, until one of them that was starting to work out was bought by [[W:Netscape|Netscape]] (for its engineering talent, not its software product) in the year of the Netscape IPO, 1995. This was a pretty good place to wind up, at the hottest start-up I had ever been early to, growing so fast it made our heads spin just to come to work the next day, at perhaps the most interesting nexus in Silicon Valley history thus far, the birth of the internet. But Netscape did not work out either in the end, as everybody knows. So I retired at the end of 1998 from writing programs other people wanted, to look for work that interested me. My principal interest had become studying symmetries in and among languages of non-deterministic computational mathematics. I had hard-won experience with computer programming languages and operating systems' internals, and a deep fascination with the problem of concurrency, but I had no mathematics. I had always disliked doing math homework assignments, the same way I detest repetitive unoriginal programming tasks.{{Efn|They are what computers are for.}} They were both hard for me for some reason.{{Efn|I think I may be algebraically dyslexic. If I don't have a picture of them, even double negatives make me work up a sweat.}} I could barely bring myself to do the grunt work my trade often demanded, and got into trouble at several companies by spending too much time making the assigned task interesting to me by biggering the design. After I failed the AP calculus exam in high school by failing to practice doing enough calculus, my formal education in mathematics ceased. At Netscape, where I was not a researcher just a programmer, and not of the original Netscape browser (Mosaic/Navigator/Mozilla), I had met the researcher [[W:Ramanathan V. Guha|R.V. Guha]], whose [[W:Meta Content Framework|Meta Content Framework (MCF)]] was in the process of being standardized as [[W:Resource Description Framework|Resource Description Framework (RDF)]], the semantic data representation that would later become a core component of Tim Berners-Lee's semantic web initiative. RDF is a language root system of semantic triples, subject-predicate-object, essentially the same data model used by the original AI researchers, who sought to construct language models by design, long before modern large language models were invented. Those are grown, not designed, from machine experience of the symmetries to be found in vast quantities of human speech found in nature. Modern AIs are not built by hand as RDF models by human architects; they are products of Darwinian natural selection, like us, rather than products of design, like the things we build. As a knowledge root system RDF interested me, but it has no operations, only a data model. After leaving Netscape I worked independently{{Efn|"Independently" in every sense: nobody paid me, I had no institutional affiliation of any kind any more, I had no colleagues any more, I published nothing and never even released any open-source code. I was independent in the sense of being completely solitary and invisible and unknowable, such that nobody could possibly have taken notice of my work even if they had been so inclined. I worked like Emily Dickinson during her lifetime, though with much less genius and prolificacy. But I was free and having fun and didn't care.}} for years programming a kind of symmetry group, consisting of the 3-dimensional RDF triples extended in a temporal 4th dimension like the one in Minkowski spacetime (the 4th field I added to the triple was a creation timestamp), and with [[W:David Gelernter|Gelernter]]'s four concurrency group operators rd(), out(), in(), and eval().{{Sfn|Carriero|Gelernter|1989|loc=The C-Linda programming language}} I realized my system as C++ template metaprograms{{Sfn|Stroustrup|2013|loc=''The [[W:C++|C++ Programming Language]]''}} for a nesting set of these operators, implemented as ACID-transactional C++ sequence iterators over the progressively more complex spaces traversed by each of Gelernter's operations. To this hierarchy I added an anonymous 5th operator()() (operator function call) between in() and eval(). None of this graph database transaction processing monitor{{Sfn|Gray|Reuter|1993|loc=''[[W:Transaction processing|Transaction processing]]''}} that I built bore any conscious resemblance to the sequence of 4-polytopes (hyper-polyhedra) of increasing complexity, with their 4th operator the 24-cell (hyper-cuboctahedron), as I had not yet resumed my college-days' study of geometry. I had studied some physics (relativity) in college and understood it as the geometry of 4-dimensional space, I had wondered about Bucky Fuller's jitterbug ever since those days, and I had been pleased to "discover" Pascal's triangle of the ''n''-simplexes back in Chicago. But I had not read much of [[W:Regular Polytopes|Regular Polytopes]] yet, and had not made the acquaintance of the more astonishing objects [[W:Ludwig Schläfli|Schläfli]] discovered in 4-dimensional Euclidean space. Particularly not yet the unique 24-cell (hyper-cuboctahedron), the radially equilateral vector equilibrium Bucky Fuller saw the cuboctahedral shadow of. Fuller searched all his life for this object (the utterly unique realization of the 24-point symmetry group of the tetrahedron), but never quite found it because he was looking for it in the wrong space (3-space). .... == Nature is symmetry == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[wikipedia:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008|loc=''The Symmetries of Things''}} As I understand [[wikipedia:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[wikipedia:Theory of relativity|Einstein's relativity]] or [[wikipedia:Evolution|Darwin's evolution]] or [[wikipedia:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of the distinct [[wikipedia:symmetry group|symmetry groups]]. .... == Poetry is symmetry == [[W:Edna St. Vincent Millay|Edna St. Vincent Millay]] and many other [[W:Lawrence Ferlinghetti#Poetry|American and non-American poets]] knew that poetry is the Insurgent Art of inventing symmetries, and at its best contains a discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, [[W:Alicia Boole Stott|Alicia Boole Stott]]. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in [[W:George Dillon (poet)|George Dillon]] the soul of an earlier Parisian youth who burned brightly, [[W:Évariste Galois|Évariste Galois]] who discovered the mathematics which underlies geometry, inventing symmetry group theory before his own fatal interview at 20. Millay's contemporary poet [[W:Emmy Noether|Emmy Noether]], the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself an expression of an exact symmetry group. These poets knew how great poetry emerges from discovery, or rediscovery, of nature's symmetries. == Justice is symmetry == Anyone should understand Israelis' unquenchable thirst for vengence for all acts that have attempted to exterminate them. Precisely because it is unquenchable, a survival instinct acquired at immeasurable cost, we must implacably resist, by all nonviolent means available to us, their attempt to slake it. Vengeance is mine, saith the Lord, and an eye for an eye makes the whole world blind. <blockquote> We are deceived into believing that we can get the kind of world we seek by doing the very things we are trying to get rid of. "Just a little more violence to end violence." "Just a little more hatred to end hatred." "Just a little more oppression to end oppression" -- and on and on. We are taken in because good people are doing these things, sincere and brave people. And this is why the finer their qualities, the more dangerous they are, the more thoroughly we are fooled. All the finest qualities in the world cannot change the simple, immutable fact that the ends cannot justify the means, but, on the contrary, the means determine the ends. In all of man's history this stands out clearly and intellectually indisputable; yet it has been perversely, insistently, sentimentally and tragically ignored. In this universe the means always and everywhere, without doubt and without exception, cannot, in the very nature of things, but determine the ends. This cannot be repeated often enough. We get what we do; not what we intend, dream, or desire. We simply get what we do. Recognizing this and applying it would, in a generation, bring about the transformation that alone can put an end to the fear, suspicion and misery which at present hold such terrible sway over all of our lives. If we see and act upon this (I will say again, unabashedly, what it is -- the means determine the ends!), then what the prophets of the ages have wistfully called Utopia will become a reality. “Nation shall no more lift sword against nation” nor unloose napalm, nerve gas or nuclear weapons. “Neither shall they learn war any more. But they shall sit every man under his vine and under his fig tree; and none shall make them afraid.” Because they will have at last understood, because we will all have at least understood, what is required of us. “To do justly and to love mercy, and to walk humbly” with the knowledge that all our means are but temporary ends and that all our ends are but new beginnings. We will have learned what every flower has never forgotten and what all oceans patiently remind us of. : - [[W:Ira Sandperl|Ira Sandperl]]{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14}} </blockquote> == Religion isn't symmetry == They say a dog is a man's best friend, but not every man should have a dog. It depends on the man. And the dog. If you want to have a dog, or a religion, as a companion and soulmate to help you answer important questions outside science's purvue, like how to be happy without making other people unhappy, have at it, and dog bless you. People who treat their religion as a source of facts about the world, instead of as a source of mystery, haven't received word yet that we have already passed through that revolutionary period in human history a few centuries back called [[w:Age_of_Enlightenment|the Enlightenment]]. Hello, we've discovered that the origin story of the facts is not mythologies, it's science. Just as you musn't let a dog drive your car or let a religion drive your government, you mustn't let a dog advise you on investment decisions or let a dogma dispute the facts that science has discovered. Religion has to stay in its lane. People who drive their religion weaving all over the road are a menace. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=[[W:Kepler's Books|Kepler's Books]]|place=Menlo Park, CA|isbn=0-8314-0035-8|jstor=73-93870|url=https://www.irasandperl.org/wordpress/index.php}} * {{Cite journal|last2=Gelernter|first2=David|author2-link=W:David Gelernter|last1=Carriero|first1=Nicholas|title=How to Write Parallel Programs: A Guide to the Perplexed|date=1989|journal=ACM Computing Surveys|volume=21|issue=3|url=https://dl.acm.org/doi/pdf/10.1145/72551.72553}} * {{Cite book|last=Stroustrup|first=Bjarne|title=The [[W:C++|C++ Programming Language]]: C++11|edition=4th|date=2013|author-link=W:Bjarne Stroustrup|publisher=Addison-Wesley}} * {{Cite book|last1=Gray|first1=Jim|author-link=W:Jim Gray (computer scientist)|last2=Reuter|first2=Andreas|title=Transaction Processing: Concepts and Techniques|title-link=W:Transaction processing|date=1993|publisher}=Morgan Kaufmann|place=San Mateo, CA}} {{Refend}} 6mwnwvm6phe9523kgkop3uvxevd9loe 2691690 2691689 2024-12-12T21:46:01Z Dc.samizdat 2856930 Undo revision [[Special:Diff/2691689|2691689]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2691690 wikitext text/x-wiki I am David Brooks Christie, born April 3, 1951. <blockquote>He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.<ref>{{Cite book|title=By His Bootstraps|last=Heinlein|first=Robert|work=The Menace From Earth|publisher=|year=1959|isbn=|location=|pages=|author-link=W:Robert Heinlein}}</ref> </blockquote> and yet... <blockquote>Oh, what good it does the heart, to know it isn't magic!<ref>{{Cite book|url=https://books.google.com/books?id=R1HTBQAAQBAJ&pg=PT43&lpg=PT43&dq=I+used+to+imagine+him+coming+from+his+house,+like+Merlin&source=bl&ots=h707RlOyPN&sig=ACfU3U22OoWoBsZwY2B-Si_jc1jdy-re1w&hl=en&sa=X&ved=2ahUKEwj06-ebwZTiAhVFvZ4KHXVkAaMQ6AEwAHoECAcQAQ#v=onepage&q=I%20used%20to%20imagine%20him%20coming%20from%20his%20house%2C%20like%20Merlin&f=false|title=Stanley Kunitz|last=Oliver|first=Mary|work=Dream Work|publisher=|year=1986|isbn=|location=|pages=|author-link=W:Mary Oliver}}</ref> </blockquote> == Polyscheme project == {{Polyscheme}} The [[Polyscheme]] project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes that pop-up in context. Some of what's in these companion articles is opinion, not established fact, as of this date of publication. == Research articles == Wiki articles I write or contribute to which contain original research or commentary, and so cannot be published as [[wikipedia:No original research|Wikipedia articles]], are hosted here at Wikiversity instead. When complete they can be submitted for peer review, and in some cases for publication elsewhere. Perhaps eventually some will even qualify to be cited as sources by Wikipedia articles. [[WikiJournal Preprints/Kinematics of the cuboctahedron]] [[WikiJournal Preprints/24-cell]] == History == I was influenced by the works of [[W:Buckminster Fuller|Buckminster Fuller]], whose ''Education Automation''<ref>{{Cite book|title=Education Automation: Comprehensive Learning for Emergent Humanity|last=Fuller|first=R. Buckminster|author-link=W:Buckminster Fuller|year=1962|editor-last=Snyder|editor-first=Jaime|publisher=Lars Müller|url=https://www.lars-mueller-publishers.com/education-automation}}</ref> I discovered in college at [[W:Oberlin college|Oberlin]]. Over the next years I read everything he published, and built a [[W:Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]] tree house centered 73.5 feet up between the twin trunks of a white pine tree in Vermont. I once walked in at the middle of one of Bucky's famous 5-hour lectures about everything, at U.C. Santa Barbara on my first trip to California in January 1971. Later I taught an Experimental College course at Oberlin on Fuller's works, where we built and flew a geodesic hot air balloon that we watched fly away to Canada over Lake Erie. We didn't think it set anything in Canada on fire with its open flame burner, because it didn't appear to start its descent until it had exhausted its fuel. After college I moved to Chicago, where I did programming in assembly language for one of the first large (nation-wide) online database transaction processing systems, hosted by a single IBM System/370 mainframe that occupied an entire floor of an office tower in the downtown Chicago loop. From Chicago I led summer-long Wilderness Projects in canoes through the Canadian taiga, straddling the 60th parallel near where the Northwest Territories, Nunavut, Manitoba and Saskatchewan meet clockwise at a four-corners, 1400 miles northwest of Chicago, 400 miles north of the northernmost town you can reach by road, and 500 miles north of our nearest radio contact, in those days before satellite phones. In 1978 I landed a programming job in California working at a tiny microcomputer manufacturer, one of the earliest such start-ups, the year after the Apple II personal computer was introduced, three years before the advent of the IBM PC. In 1982 I reached Silicon Valley, the end of the rainbow for ambitious programmers, epicenter of the emergence, swiftly urbanizing frontier homestead of the digital revolution. We lived in a refurbished cabin in a redwood forest on the wild San Mateo County coast, and I commuted over-the-hill every day to my job at a computer manufacturer, in the same truck I had commuted to Canada in with 3 canoes on the roof every summer from Chicago. At [[W:Convergent Technologies|that company]], the job I held longest in my peripatetic career (7 years) and learned the most from, they built the first fully concurrent, ethernet-networked office workstations from Intel's newest microprocessors (before the silicon was even dry and fully debugged), and I wrote code for the internals of their distributed, message-based operating system. At last I had arrived at a long-sought destination, the root systems of the computer. In Silicon Valley I worked for stock options at a series of interesting start-ups that did not work out, until one of them that was starting to work out was bought by [[W:Netscape|Netscape]] (for its engineering talent, not its software product) in the year of the Netscape IPO, 1995. This was a pretty good place to wind up, at the hottest start-up I had ever been early to, growing so fast it made our heads spin just to come to work the next day, at perhaps the most interesting nexus in Silicon Valley history thus far, the birth of the internet. But Netscape did not work out either in the end, as everybody knows. So I retired at the end of 1998 from writing programs other people wanted, to look for work that interested me. My principal interest had become studying symmetries in and among languages of non-deterministic computational mathematics. I had hard-won experience with computer programming languages and operating systems' internals, and a deep fascination with the problem of concurrency, but I had no mathematics. I had always disliked doing math homework assignments, the same way I detest repetitive unoriginal programming tasks.{{Efn|They are what computers are for.}} They were both hard for me for some reason.{{Efn|I think I may be algebraically dyslexic. If I don't have a picture of them, even double negatives make me work up a sweat.}} I could barely bring myself to do the grunt work my trade often demanded, and got into trouble at several companies by spending too much time making the assigned task interesting to me by biggering the design. After I failed the AP calculus exam in high school by failing to practice doing enough calculus, my formal education in mathematics ceased. At Netscape, where I was not a researcher just a programmer, and not of the original Netscape browser (Mosaic/Navigator/Mozilla), I had met the researcher [[W:Ramanathan V. Guha|R.V. Guha]], whose [[W:Meta Content Framework|Meta Content Framework (MCF)]] was in the process of being standardized as [[W:Resource Description Framework|Resource Description Framework (RDF)]], the semantic data representation that would later become a core component of Tim Berners-Lee's semantic web initiative. RDF is a language root system of semantic triples, subject-predicate-object, essentially the same data model used by the original AI researchers, who sought to construct language models by design, long before modern large language models were invented. Those are grown, not designed, from machine experience of the symmetries to be found in vast quantities of human speech found in nature. Modern AIs are not built by hand as RDF models by human architects; they are products of Darwinian natural selection, like us, rather than products of design, like the things we build. As a knowledge root system RDF interested me, but it has no operations, only a data model. After leaving Netscape I worked independently{{Efn|"Independently" in every sense: nobody paid me, I had no institutional affiliation of any kind any more, I had no colleagues any more, I published nothing and never even released any open-source code. I was independent in the sense of being completely solitary and invisible and unknowable, such that nobody could possibly have taken notice of my work even if they had been so inclined. I worked like Emily Dickinson during her lifetime, though with much less genius and prolificacy. But I was free and having fun and didn't care.}} for years programming a kind of symmetry group, consisting of the 3-dimensional RDF triples extended in a temporal 4th dimension like the one in Minkowski spacetime (the 4th field I added to the triple was a creation timestamp), and with [[W:David Gelernter|Gelernter]]'s four concurrency group operators rd(), out(), in(), and eval().{{Sfn|Carriero|Gelernter|1989|loc=The C-Linda programming language}} I realized my system as C++ template metaprograms{{Sfn|Stroustrup|2013|loc=''The [[W:C++|C++ Programming Language]]''}} for a nesting set of these operators, implemented as ACID-transactional C++ sequence iterators over the progressively more complex spaces traversed by each of Gelernter's operations. To this hierarchy I added an anonymous 5th operator()() (operator function call) between in() and eval(). None of this graph database transaction processing monitor{{Sfn|Gray|Reuter|1993|loc=''[[W:Transaction processing|Transaction processing]]''}} that I built bore any conscious resemblance to the sequence of 4-polytopes (hyper-polyhedra) of increasing complexity, with their 4th operator the 24-cell (hyper-anonymous), as I had not yet resumed my college-days' study of geometry. I had studied some physics (relativity) in college and understood it as the geometry of 4-dimensional space, I had wondered about Bucky Fuller's jitterbug ever since those days, and I had been pleased to "discover" Pascal's triangle of the ''n''-simplexes back in Chicago. But I had not read much of [[W:Regular Polytopes|Regular Polytopes]] yet, and had not made the acquaintance of the more astonishing objects [[W:Ludwig Schläfli|Schläfli]] discovered in 4-dimensional Euclidean space. Particularly not yet the unique 24-cell (hyper-cuboctahedron), the radially equilateral vector equilibrium Bucky Fuller saw the cuboctahedral shadow of. Fuller searched all his life for this object (the utterly unique realization of the 24-point symmetry group of the tetrahedron), but never quite found it because he was looking for it in the wrong space (3-space). .... == Nature is symmetry == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[wikipedia:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008|loc=''The Symmetries of Things''}} As I understand [[wikipedia:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[wikipedia:Theory of relativity|Einstein's relativity]] or [[wikipedia:Evolution|Darwin's evolution]] or [[wikipedia:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of the distinct [[wikipedia:symmetry group|symmetry groups]]. .... == Poetry is symmetry == [[W:Edna St. Vincent Millay|Edna St. Vincent Millay]] and many other [[W:Lawrence Ferlinghetti#Poetry|American and non-American poets]] knew that poetry is the Insurgent Art of inventing symmetries, and at its best contains a discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, [[W:Alicia Boole Stott|Alicia Boole Stott]]. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in [[W:George Dillon (poet)|George Dillon]] the soul of an earlier Parisian youth who burned brightly, [[W:Évariste Galois|Évariste Galois]] who discovered the mathematics which underlies geometry, inventing symmetry group theory before his own fatal interview at 20. Millay's contemporary poet [[W:Emmy Noether|Emmy Noether]], the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself an expression of an exact symmetry group. These poets knew how great poetry emerges from discovery, or rediscovery, of nature's symmetries. == Justice is symmetry == Anyone should understand Israelis' unquenchable thirst for vengence for all acts that have attempted to exterminate them. Precisely because it is unquenchable, a survival instinct acquired at immeasurable cost, we must implacably resist, by all nonviolent means available to us, their attempt to slake it. Vengeance is mine, saith the Lord, and an eye for an eye makes the whole world blind. <blockquote> We are deceived into believing that we can get the kind of world we seek by doing the very things we are trying to get rid of. "Just a little more violence to end violence." "Just a little more hatred to end hatred." "Just a little more oppression to end oppression" -- and on and on. We are taken in because good people are doing these things, sincere and brave people. And this is why the finer their qualities, the more dangerous they are, the more thoroughly we are fooled. All the finest qualities in the world cannot change the simple, immutable fact that the ends cannot justify the means, but, on the contrary, the means determine the ends. In all of man's history this stands out clearly and intellectually indisputable; yet it has been perversely, insistently, sentimentally and tragically ignored. In this universe the means always and everywhere, without doubt and without exception, cannot, in the very nature of things, but determine the ends. This cannot be repeated often enough. We get what we do; not what we intend, dream, or desire. We simply get what we do. Recognizing this and applying it would, in a generation, bring about the transformation that alone can put an end to the fear, suspicion and misery which at present hold such terrible sway over all of our lives. If we see and act upon this (I will say again, unabashedly, what it is -- the means determine the ends!), then what the prophets of the ages have wistfully called Utopia will become a reality. “Nation shall no more lift sword against nation” nor unloose napalm, nerve gas or nuclear weapons. “Neither shall they learn war any more. But they shall sit every man under his vine and under his fig tree; and none shall make them afraid.” Because they will have at last understood, because we will all have at least understood, what is required of us. “To do justly and to love mercy, and to walk humbly” with the knowledge that all our means are but temporary ends and that all our ends are but new beginnings. We will have learned what every flower has never forgotten and what all oceans patiently remind us of. : - [[W:Ira Sandperl|Ira Sandperl]]{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14}} </blockquote> == Religion isn't symmetry == They say a dog is a man's best friend, but not every man should have a dog. It depends on the man. And the dog. If you want to have a dog, or a religion, as a companion and soulmate to help you answer important questions outside science's purvue, like how to be happy without making other people unhappy, have at it, and dog bless you. People who treat their religion as a source of facts about the world, instead of as a source of mystery, haven't received word yet that we have already passed through that revolutionary period in human history a few centuries back called [[w:Age_of_Enlightenment|the Enlightenment]]. Hello, we've discovered that the origin story of the facts is not mythologies, it's science. Just as you musn't let a dog drive your car or let a religion drive your government, you mustn't let a dog advise you on investment decisions or let a dogma dispute the facts that science has discovered. Religion has to stay in its lane. People who drive their religion weaving all over the road are a menace. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=[[W:Kepler's Books|Kepler's Books]]|place=Menlo Park, CA|isbn=0-8314-0035-8|jstor=73-93870|url=https://www.irasandperl.org/wordpress/index.php}} * {{Cite journal|last2=Gelernter|first2=David|author2-link=W:David Gelernter|last1=Carriero|first1=Nicholas|title=How to Write Parallel Programs: A Guide to the Perplexed|date=1989|journal=ACM Computing Surveys|volume=21|issue=3|url=https://dl.acm.org/doi/pdf/10.1145/72551.72553}} * {{Cite book|last=Stroustrup|first=Bjarne|title=The [[W:C++|C++ Programming Language]]: C++11|edition=4th|date=2013|author-link=W:Bjarne Stroustrup|publisher=Addison-Wesley}} * {{Cite book|last1=Gray|first1=Jim|author-link=W:Jim Gray (computer scientist)|last2=Reuter|first2=Andreas|title=Transaction Processing: Concepts and Techniques|title-link=W:Transaction processing|date=1993|publisher}=Morgan Kaufmann|place=San Mateo, CA}} {{Refend}} e40m0pmlbpowajxsr4uao495wqfmbwv 2691691 2691690 2024-12-12T21:55:05Z Dc.samizdat 2856930 /* History */ 2691691 wikitext text/x-wiki I am David Brooks Christie, born April 3, 1951. <blockquote>He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.<ref>{{Cite book|title=By His Bootstraps|last=Heinlein|first=Robert|work=The Menace From Earth|publisher=|year=1959|isbn=|location=|pages=|author-link=W:Robert Heinlein}}</ref> </blockquote> and yet... <blockquote>Oh, what good it does the heart, to know it isn't magic!<ref>{{Cite book|url=https://books.google.com/books?id=R1HTBQAAQBAJ&pg=PT43&lpg=PT43&dq=I+used+to+imagine+him+coming+from+his+house,+like+Merlin&source=bl&ots=h707RlOyPN&sig=ACfU3U22OoWoBsZwY2B-Si_jc1jdy-re1w&hl=en&sa=X&ved=2ahUKEwj06-ebwZTiAhVFvZ4KHXVkAaMQ6AEwAHoECAcQAQ#v=onepage&q=I%20used%20to%20imagine%20him%20coming%20from%20his%20house%2C%20like%20Merlin&f=false|title=Stanley Kunitz|last=Oliver|first=Mary|work=Dream Work|publisher=|year=1986|isbn=|location=|pages=|author-link=W:Mary Oliver}}</ref> </blockquote> == Polyscheme project == {{Polyscheme}} The [[Polyscheme]] project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes that pop-up in context. Some of what's in these companion articles is opinion, not established fact, as of this date of publication. == Research articles == Wiki articles I write or contribute to which contain original research or commentary, and so cannot be published as [[wikipedia:No original research|Wikipedia articles]], are hosted here at Wikiversity instead. When complete they can be submitted for peer review, and in some cases for publication elsewhere. Perhaps eventually some will even qualify to be cited as sources by Wikipedia articles. [[WikiJournal Preprints/Kinematics of the cuboctahedron]] [[WikiJournal Preprints/24-cell]] == History == I was influenced by the works of [[W:Buckminster Fuller|Buckminster Fuller]], whose ''Education Automation''<ref>{{Cite book|title=Education Automation: Comprehensive Learning for Emergent Humanity|last=Fuller|first=R. Buckminster|author-link=W:Buckminster Fuller|year=1962|editor-last=Snyder|editor-first=Jaime|publisher=Lars Müller|url=https://www.lars-mueller-publishers.com/education-automation}}</ref> I discovered in college at [[W:Oberlin college|Oberlin]]. Over the next years I read everything he published, and built a [[W:Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]] tree house centered 73.5 feet up between the twin trunks of a white pine tree in Vermont. I once walked in at the middle of one of Bucky's famous 5-hour lectures about everything, at U.C. Santa Barbara on my first trip to California in January 1971. Later I taught an Experimental College course at Oberlin on Fuller's works, where we built and flew a geodesic hot air balloon that we watched fly away to Canada over Lake Erie. We didn't think it set anything in Canada on fire with its open flame burner, because it didn't appear to start its descent until it had exhausted its fuel. After college I moved to Chicago, where I did programming in assembly language for one of the first large (nation-wide) online database transaction processing systems, hosted by a single IBM System/370 mainframe that occupied an entire floor of an office tower in the downtown Chicago loop. From Chicago I led summer-long Wilderness Projects in canoes through the Canadian taiga, straddling the 60th parallel near where the Northwest Territories, Nunavut, Manitoba and Saskatchewan meet clockwise at a four-corners, 1400 miles northwest of Chicago, 400 miles north of the northernmost town you can reach by road, and 500 miles north of our nearest radio contact, in those days before satellite phones. In 1978 I landed a programming job in California working at a tiny microcomputer manufacturer, one of the earliest such start-ups, the year after the Apple II personal computer was introduced, three years before the advent of the IBM PC. In 1982 I reached Silicon Valley, the end of the rainbow for ambitious programmers, epicenter of the emergence, swiftly urbanizing frontier homestead of the digital revolution. We lived in a refurbished cabin in a redwood forest on the wild San Mateo County coast, and I commuted over-the-hill every day to my job at a computer manufacturer, in the same truck I had commuted to Canada in with 3 canoes on the roof every summer from Chicago. At [[W:Convergent Technologies|that company]], the job I held longest in my peripatetic career (7 years) and learned the most from, they built the first fully concurrent, ethernet-networked office workstations from Intel's newest microprocessors (before the silicon was even dry and fully debugged), and I wrote code for the internals of their distributed, message-based operating system. At last I had arrived at a long-sought destination, the root systems of the computer. In Silicon Valley I worked for stock options at a series of interesting start-ups that did not work out, until one of them that was starting to work out was bought by [[W:Netscape|Netscape]] (for its engineering talent, not its software product) in the year of the Netscape IPO, 1995. This was a pretty good place to wind up, at the hottest start-up I had ever been early to, growing so fast it made our heads spin just to come to work the next day, at perhaps the most interesting nexus in Silicon Valley history thus far, the birth of the internet. But Netscape did not work out either in the end, as everybody knows. So I retired at the end of 1998 from writing programs other people wanted, to look for work that interested me. My principal interest had become studying symmetries in and among languages of non-deterministic computational mathematics. I had hard-won experience with computer programming languages and operating systems' internals, and a deep fascination with the problem of concurrency, but I had no mathematics. I had always disliked doing math homework assignments, the same way I detest repetitive unoriginal programming tasks.{{Efn|They are what computers are for.}} They were both hard for me for some reason.{{Efn|I think I may be algebraically dyslexic. If I don't have a picture of them, even double negatives make me work up a sweat.}} I could barely bring myself to do the grunt work my trade often demanded, and got into trouble at several companies by spending too much time making the assigned task interesting to me by biggering the design. After I failed the AP calculus exam in high school by failing to practice doing enough calculus, my formal education in mathematics ceased. I am not proud of being mathematically illiterate. I am severely inconvenienced by it in my work, but I have adapted to my disability. At Netscape, where I was not a researcher just a programmer, and not of the original Netscape browser (Mosaic/Navigator/Mozilla), I had met the researcher [[W:Ramanathan V. Guha|R.V. Guha]], whose [[W:Meta Content Framework|Meta Content Framework (MCF)]] was in the process of being standardized as [[W:Resource Description Framework|Resource Description Framework (RDF)]], the semantic data representation that would later become a core component of Tim Berners-Lee's semantic web initiative. RDF is a language root system of semantic triples, subject-predicate-object, essentially the same data model used by the original AI researchers, who sought to construct language models by design, long before modern large language models were invented. Those are grown, not designed, from machine experience of the symmetries to be found in vast quantities of human speech found in nature. Modern AIs are not built by hand as RDF models by human architects; they are products of Darwinian natural selection, like us, rather than products of design, like the things we build. As a knowledge root system RDF interested me, but it has no operations, only a data model. After leaving Netscape I worked independently{{Efn|"Independently" in every sense: nobody paid me, I had no institutional affiliation of any kind any more, I had no colleagues any more, I published nothing and never even released any open-source code. I was independent in the sense of being completely solitary and invisible and unknowable, such that nobody could possibly have taken notice of my work even if they had been so inclined. I worked like Emily Dickinson during her lifetime, though with much less genius and prolificacy. But I was free and having fun and didn't care.}} for years programming a kind of symmetry group, consisting of the 3-dimensional RDF triples extended in a temporal 4th dimension like the one in Minkowski spacetime (the 4th field I added to the triple was a creation timestamp), and with [[W:David Gelernter|Gelernter]]'s four concurrency group operators rd(), out(), in(), and eval().{{Sfn|Carriero|Gelernter|1989|loc=The C-Linda programming language}} I realized my system as C++ template metaprograms{{Sfn|Stroustrup|2013|loc=''The [[W:C++|C++ Programming Language]]''}} for a nesting set of these operators, implemented as ACID-transactional C++ sequence iterators over the progressively more complex spaces traversed by each of Gelernter's operations. To this hierarchy I added an anonymous 5th operator()() (operator function call) between in() and eval(). None of this graph database transaction processing monitor{{Sfn|Gray|Reuter|1993|loc=''[[W:Transaction processing|Transaction processing]]''}} that I built bore any conscious resemblance to the sequence of 4-polytopes (hyper-polyhedra) of increasing complexity, with their 4th operator the 24-cell (hyper-anonymous), as I had not yet resumed my college-days' study of geometry. I had studied some physics (relativity) in college and understood it as the geometry of 4-dimensional space, I had wondered about Bucky Fuller's jitterbug ever since those days, and I had been pleased to "discover" Pascal's triangle of the ''n''-simplexes back in Chicago. But I had not read much of [[W:Regular Polytopes|Regular Polytopes]] yet, and had not made the acquaintance of the more astonishing objects [[W:Ludwig Schläfli|Schläfli]] discovered in 4-dimensional Euclidean space. Particularly not yet the unique 24-cell (hyper-cuboctahedron), the radially equilateral vector equilibrium Bucky Fuller saw the cuboctahedral shadow of. Fuller searched all his life for this object (the utterly unique realization of the 24-point symmetry group of the tetrahedron), but never quite found it because he was looking for it in the wrong space (3-space). .... == Nature is symmetry == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[wikipedia:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008|loc=''The Symmetries of Things''}} As I understand [[wikipedia:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[wikipedia:Theory of relativity|Einstein's relativity]] or [[wikipedia:Evolution|Darwin's evolution]] or [[wikipedia:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of the distinct [[wikipedia:symmetry group|symmetry groups]]. .... == Poetry is symmetry == [[W:Edna St. Vincent Millay|Edna St. Vincent Millay]] and many other [[W:Lawrence Ferlinghetti#Poetry|American and non-American poets]] knew that poetry is the Insurgent Art of inventing symmetries, and at its best contains a discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, [[W:Alicia Boole Stott|Alicia Boole Stott]]. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in [[W:George Dillon (poet)|George Dillon]] the soul of an earlier Parisian youth who burned brightly, [[W:Évariste Galois|Évariste Galois]] who discovered the mathematics which underlies geometry, inventing symmetry group theory before his own fatal interview at 20. Millay's contemporary poet [[W:Emmy Noether|Emmy Noether]], the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself an expression of an exact symmetry group. These poets knew how great poetry emerges from discovery, or rediscovery, of nature's symmetries. == Justice is symmetry == Anyone should understand Israelis' unquenchable thirst for vengence for all acts that have attempted to exterminate them. Precisely because it is unquenchable, a survival instinct acquired at immeasurable cost, we must implacably resist, by all nonviolent means available to us, their attempt to slake it. Vengeance is mine, saith the Lord, and an eye for an eye makes the whole world blind. <blockquote> We are deceived into believing that we can get the kind of world we seek by doing the very things we are trying to get rid of. "Just a little more violence to end violence." "Just a little more hatred to end hatred." "Just a little more oppression to end oppression" -- and on and on. We are taken in because good people are doing these things, sincere and brave people. And this is why the finer their qualities, the more dangerous they are, the more thoroughly we are fooled. All the finest qualities in the world cannot change the simple, immutable fact that the ends cannot justify the means, but, on the contrary, the means determine the ends. In all of man's history this stands out clearly and intellectually indisputable; yet it has been perversely, insistently, sentimentally and tragically ignored. In this universe the means always and everywhere, without doubt and without exception, cannot, in the very nature of things, but determine the ends. This cannot be repeated often enough. We get what we do; not what we intend, dream, or desire. We simply get what we do. Recognizing this and applying it would, in a generation, bring about the transformation that alone can put an end to the fear, suspicion and misery which at present hold such terrible sway over all of our lives. If we see and act upon this (I will say again, unabashedly, what it is -- the means determine the ends!), then what the prophets of the ages have wistfully called Utopia will become a reality. “Nation shall no more lift sword against nation” nor unloose napalm, nerve gas or nuclear weapons. “Neither shall they learn war any more. But they shall sit every man under his vine and under his fig tree; and none shall make them afraid.” Because they will have at last understood, because we will all have at least understood, what is required of us. “To do justly and to love mercy, and to walk humbly” with the knowledge that all our means are but temporary ends and that all our ends are but new beginnings. We will have learned what every flower has never forgotten and what all oceans patiently remind us of. : - [[W:Ira Sandperl|Ira Sandperl]]{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14}} </blockquote> == Religion isn't symmetry == They say a dog is a man's best friend, but not every man should have a dog. It depends on the man. And the dog. If you want to have a dog, or a religion, as a companion and soulmate to help you answer important questions outside science's purvue, like how to be happy without making other people unhappy, have at it, and dog bless you. People who treat their religion as a source of facts about the world, instead of as a source of mystery, haven't received word yet that we have already passed through that revolutionary period in human history a few centuries back called [[w:Age_of_Enlightenment|the Enlightenment]]. Hello, we've discovered that the origin story of the facts is not mythologies, it's science. Just as you musn't let a dog drive your car or let a religion drive your government, you mustn't let a dog advise you on investment decisions or let a dogma dispute the facts that science has discovered. Religion has to stay in its lane. People who drive their religion weaving all over the road are a menace. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=[[W:Kepler's Books|Kepler's Books]]|place=Menlo Park, CA|isbn=0-8314-0035-8|jstor=73-93870|url=https://www.irasandperl.org/wordpress/index.php}} * {{Cite journal|last2=Gelernter|first2=David|author2-link=W:David Gelernter|last1=Carriero|first1=Nicholas|title=How to Write Parallel Programs: A Guide to the Perplexed|date=1989|journal=ACM Computing Surveys|volume=21|issue=3|url=https://dl.acm.org/doi/pdf/10.1145/72551.72553}} * {{Cite book|last=Stroustrup|first=Bjarne|title=The [[W:C++|C++ Programming Language]]: C++11|edition=4th|date=2013|author-link=W:Bjarne Stroustrup|publisher=Addison-Wesley}} * {{Cite book|last1=Gray|first1=Jim|author-link=W:Jim Gray (computer scientist)|last2=Reuter|first2=Andreas|title=Transaction Processing: Concepts and Techniques|title-link=W:Transaction processing|date=1993|publisher}=Morgan Kaufmann|place=San Mateo, CA}} {{Refend}} 3e0iio98oq8h4cfvg00vare0ot6h7go 2691701 2691691 2024-12-12T22:29:57Z Dc.samizdat 2856930 /* Nature is symmetry */ 2691701 wikitext text/x-wiki I am David Brooks Christie, born April 3, 1951. <blockquote>He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.<ref>{{Cite book|title=By His Bootstraps|last=Heinlein|first=Robert|work=The Menace From Earth|publisher=|year=1959|isbn=|location=|pages=|author-link=W:Robert Heinlein}}</ref> </blockquote> and yet... <blockquote>Oh, what good it does the heart, to know it isn't magic!<ref>{{Cite book|url=https://books.google.com/books?id=R1HTBQAAQBAJ&pg=PT43&lpg=PT43&dq=I+used+to+imagine+him+coming+from+his+house,+like+Merlin&source=bl&ots=h707RlOyPN&sig=ACfU3U22OoWoBsZwY2B-Si_jc1jdy-re1w&hl=en&sa=X&ved=2ahUKEwj06-ebwZTiAhVFvZ4KHXVkAaMQ6AEwAHoECAcQAQ#v=onepage&q=I%20used%20to%20imagine%20him%20coming%20from%20his%20house%2C%20like%20Merlin&f=false|title=Stanley Kunitz|last=Oliver|first=Mary|work=Dream Work|publisher=|year=1986|isbn=|location=|pages=|author-link=W:Mary Oliver}}</ref> </blockquote> == Polyscheme project == {{Polyscheme}} The [[Polyscheme]] project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes that pop-up in context. Some of what's in these companion articles is opinion, not established fact, as of this date of publication. == Research articles == Wiki articles I write or contribute to which contain original research or commentary, and so cannot be published as [[wikipedia:No original research|Wikipedia articles]], are hosted here at Wikiversity instead. When complete they can be submitted for peer review, and in some cases for publication elsewhere. Perhaps eventually some will even qualify to be cited as sources by Wikipedia articles. [[WikiJournal Preprints/Kinematics of the cuboctahedron]] [[WikiJournal Preprints/24-cell]] == History == I was influenced by the works of [[W:Buckminster Fuller|Buckminster Fuller]], whose ''Education Automation''<ref>{{Cite book|title=Education Automation: Comprehensive Learning for Emergent Humanity|last=Fuller|first=R. Buckminster|author-link=W:Buckminster Fuller|year=1962|editor-last=Snyder|editor-first=Jaime|publisher=Lars Müller|url=https://www.lars-mueller-publishers.com/education-automation}}</ref> I discovered in college at [[W:Oberlin college|Oberlin]]. Over the next years I read everything he published, and built a [[W:Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]] tree house centered 73.5 feet up between the twin trunks of a white pine tree in Vermont. I once walked in at the middle of one of Bucky's famous 5-hour lectures about everything, at U.C. Santa Barbara on my first trip to California in January 1971. Later I taught an Experimental College course at Oberlin on Fuller's works, where we built and flew a geodesic hot air balloon that we watched fly away to Canada over Lake Erie. We didn't think it set anything in Canada on fire with its open flame burner, because it didn't appear to start its descent until it had exhausted its fuel. After college I moved to Chicago, where I did programming in assembly language for one of the first large (nation-wide) online database transaction processing systems, hosted by a single IBM System/370 mainframe that occupied an entire floor of an office tower in the downtown Chicago loop. From Chicago I led summer-long Wilderness Projects in canoes through the Canadian taiga, straddling the 60th parallel near where the Northwest Territories, Nunavut, Manitoba and Saskatchewan meet clockwise at a four-corners, 1400 miles northwest of Chicago, 400 miles north of the northernmost town you can reach by road, and 500 miles north of our nearest radio contact, in those days before satellite phones. In 1978 I landed a programming job in California working at a tiny microcomputer manufacturer, one of the earliest such start-ups, the year after the Apple II personal computer was introduced, three years before the advent of the IBM PC. In 1982 I reached Silicon Valley, the end of the rainbow for ambitious programmers, epicenter of the emergence, swiftly urbanizing frontier homestead of the digital revolution. We lived in a refurbished cabin in a redwood forest on the wild San Mateo County coast, and I commuted over-the-hill every day to my job at a computer manufacturer, in the same truck I had commuted to Canada in with 3 canoes on the roof every summer from Chicago. At [[W:Convergent Technologies|that company]], the job I held longest in my peripatetic career (7 years) and learned the most from, they built the first fully concurrent, ethernet-networked office workstations from Intel's newest microprocessors (before the silicon was even dry and fully debugged), and I wrote code for the internals of their distributed, message-based operating system. At last I had arrived at a long-sought destination, the root systems of the computer. In Silicon Valley I worked for stock options at a series of interesting start-ups that did not work out, until one of them that was starting to work out was bought by [[W:Netscape|Netscape]] (for its engineering talent, not its software product) in the year of the Netscape IPO, 1995. This was a pretty good place to wind up, at the hottest start-up I had ever been early to, growing so fast it made our heads spin just to come to work the next day, at perhaps the most interesting nexus in Silicon Valley history thus far, the birth of the internet. But Netscape did not work out either in the end, as everybody knows. So I retired at the end of 1998 from writing programs other people wanted, to look for work that interested me. My principal interest had become studying symmetries in and among languages of non-deterministic computational mathematics. I had hard-won experience with computer programming languages and operating systems' internals, and a deep fascination with the problem of concurrency, but I had no mathematics. I had always disliked doing math homework assignments, the same way I detest repetitive unoriginal programming tasks.{{Efn|They are what computers are for.}} They were both hard for me for some reason.{{Efn|I think I may be algebraically dyslexic. If I don't have a picture of them, even double negatives make me work up a sweat.}} I could barely bring myself to do the grunt work my trade often demanded, and got into trouble at several companies by spending too much time making the assigned task interesting to me by biggering the design. After I failed the AP calculus exam in high school by failing to practice doing enough calculus, my formal education in mathematics ceased. I am not proud of being mathematically illiterate. I am severely inconvenienced by it in my work, but I have adapted to my disability. At Netscape, where I was not a researcher just a programmer, and not of the original Netscape browser (Mosaic/Navigator/Mozilla), I had met the researcher [[W:Ramanathan V. Guha|R.V. Guha]], whose [[W:Meta Content Framework|Meta Content Framework (MCF)]] was in the process of being standardized as [[W:Resource Description Framework|Resource Description Framework (RDF)]], the semantic data representation that would later become a core component of Tim Berners-Lee's semantic web initiative. RDF is a language root system of semantic triples, subject-predicate-object, essentially the same data model used by the original AI researchers, who sought to construct language models by design, long before modern large language models were invented. Those are grown, not designed, from machine experience of the symmetries to be found in vast quantities of human speech found in nature. Modern AIs are not built by hand as RDF models by human architects; they are products of Darwinian natural selection, like us, rather than products of design, like the things we build. As a knowledge root system RDF interested me, but it has no operations, only a data model. After leaving Netscape I worked independently{{Efn|"Independently" in every sense: nobody paid me, I had no institutional affiliation of any kind any more, I had no colleagues any more, I published nothing and never even released any open-source code. I was independent in the sense of being completely solitary and invisible and unknowable, such that nobody could possibly have taken notice of my work even if they had been so inclined. I worked like Emily Dickinson during her lifetime, though with much less genius and prolificacy. But I was free and having fun and didn't care.}} for years programming a kind of symmetry group, consisting of the 3-dimensional RDF triples extended in a temporal 4th dimension like the one in Minkowski spacetime (the 4th field I added to the triple was a creation timestamp), and with [[W:David Gelernter|Gelernter]]'s four concurrency group operators rd(), out(), in(), and eval().{{Sfn|Carriero|Gelernter|1989|loc=The C-Linda programming language}} I realized my system as C++ template metaprograms{{Sfn|Stroustrup|2013|loc=''The [[W:C++|C++ Programming Language]]''}} for a nesting set of these operators, implemented as ACID-transactional C++ sequence iterators over the progressively more complex spaces traversed by each of Gelernter's operations. To this hierarchy I added an anonymous 5th operator()() (operator function call) between in() and eval(). None of this graph database transaction processing monitor{{Sfn|Gray|Reuter|1993|loc=''[[W:Transaction processing|Transaction processing]]''}} that I built bore any conscious resemblance to the sequence of 4-polytopes (hyper-polyhedra) of increasing complexity, with their 4th operator the 24-cell (hyper-anonymous), as I had not yet resumed my college-days' study of geometry. I had studied some physics (relativity) in college and understood it as the geometry of 4-dimensional space, I had wondered about Bucky Fuller's jitterbug ever since those days, and I had been pleased to "discover" Pascal's triangle of the ''n''-simplexes back in Chicago. But I had not read much of [[W:Regular Polytopes|Regular Polytopes]] yet, and had not made the acquaintance of the more astonishing objects [[W:Ludwig Schläfli|Schläfli]] discovered in 4-dimensional Euclidean space. Particularly not yet the unique 24-cell (hyper-cuboctahedron), the radially equilateral vector equilibrium Bucky Fuller saw the cuboctahedral shadow of. Fuller searched all his life for this object (the utterly unique realization of the 24-point symmetry group of the tetrahedron), but never quite found it because he was looking for it in the wrong space (3-space). .... == Nature is symmetry == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[wikipedia:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008|loc=''The Symmetries of Things''}} As I understand [[wikipedia:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[wikipedia:Theory of relativity|Einstein's relativity]] or [[wikipedia:Evolution|Darwin's evolution]] or [[wikipedia:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[wikipedia:symmetry group|symmetry groups]]. .... == Poetry is symmetry == [[W:Edna St. Vincent Millay|Edna St. Vincent Millay]] and many other [[W:Lawrence Ferlinghetti#Poetry|American and non-American poets]] knew that poetry is the Insurgent Art of inventing symmetries, and at its best contains a discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, [[W:Alicia Boole Stott|Alicia Boole Stott]]. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in [[W:George Dillon (poet)|George Dillon]] the soul of an earlier Parisian youth who burned brightly, [[W:Évariste Galois|Évariste Galois]] who discovered the mathematics which underlies geometry, inventing symmetry group theory before his own fatal interview at 20. Millay's contemporary poet [[W:Emmy Noether|Emmy Noether]], the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself an expression of an exact symmetry group. These poets knew how great poetry emerges from discovery, or rediscovery, of nature's symmetries. == Justice is symmetry == Anyone should understand Israelis' unquenchable thirst for vengence for all acts that have attempted to exterminate them. Precisely because it is unquenchable, a survival instinct acquired at immeasurable cost, we must implacably resist, by all nonviolent means available to us, their attempt to slake it. Vengeance is mine, saith the Lord, and an eye for an eye makes the whole world blind. <blockquote> We are deceived into believing that we can get the kind of world we seek by doing the very things we are trying to get rid of. "Just a little more violence to end violence." "Just a little more hatred to end hatred." "Just a little more oppression to end oppression" -- and on and on. We are taken in because good people are doing these things, sincere and brave people. And this is why the finer their qualities, the more dangerous they are, the more thoroughly we are fooled. All the finest qualities in the world cannot change the simple, immutable fact that the ends cannot justify the means, but, on the contrary, the means determine the ends. In all of man's history this stands out clearly and intellectually indisputable; yet it has been perversely, insistently, sentimentally and tragically ignored. In this universe the means always and everywhere, without doubt and without exception, cannot, in the very nature of things, but determine the ends. This cannot be repeated often enough. We get what we do; not what we intend, dream, or desire. We simply get what we do. Recognizing this and applying it would, in a generation, bring about the transformation that alone can put an end to the fear, suspicion and misery which at present hold such terrible sway over all of our lives. If we see and act upon this (I will say again, unabashedly, what it is -- the means determine the ends!), then what the prophets of the ages have wistfully called Utopia will become a reality. “Nation shall no more lift sword against nation” nor unloose napalm, nerve gas or nuclear weapons. “Neither shall they learn war any more. But they shall sit every man under his vine and under his fig tree; and none shall make them afraid.” Because they will have at last understood, because we will all have at least understood, what is required of us. “To do justly and to love mercy, and to walk humbly” with the knowledge that all our means are but temporary ends and that all our ends are but new beginnings. We will have learned what every flower has never forgotten and what all oceans patiently remind us of. : - [[W:Ira Sandperl|Ira Sandperl]]{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14}} </blockquote> == Religion isn't symmetry == They say a dog is a man's best friend, but not every man should have a dog. It depends on the man. And the dog. If you want to have a dog, or a religion, as a companion and soulmate to help you answer important questions outside science's purvue, like how to be happy without making other people unhappy, have at it, and dog bless you. People who treat their religion as a source of facts about the world, instead of as a source of mystery, haven't received word yet that we have already passed through that revolutionary period in human history a few centuries back called [[w:Age_of_Enlightenment|the Enlightenment]]. Hello, we've discovered that the origin story of the facts is not mythologies, it's science. Just as you musn't let a dog drive your car or let a religion drive your government, you mustn't let a dog advise you on investment decisions or let a dogma dispute the facts that science has discovered. Religion has to stay in its lane. People who drive their religion weaving all over the road are a menace. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=[[W:Kepler's Books|Kepler's Books]]|place=Menlo Park, CA|isbn=0-8314-0035-8|jstor=73-93870|url=https://www.irasandperl.org/wordpress/index.php}} * {{Cite journal|last2=Gelernter|first2=David|author2-link=W:David Gelernter|last1=Carriero|first1=Nicholas|title=How to Write Parallel Programs: A Guide to the Perplexed|date=1989|journal=ACM Computing Surveys|volume=21|issue=3|url=https://dl.acm.org/doi/pdf/10.1145/72551.72553}} * {{Cite book|last=Stroustrup|first=Bjarne|title=The [[W:C++|C++ Programming Language]]: C++11|edition=4th|date=2013|author-link=W:Bjarne Stroustrup|publisher=Addison-Wesley}} * {{Cite book|last1=Gray|first1=Jim|author-link=W:Jim Gray (computer scientist)|last2=Reuter|first2=Andreas|title=Transaction Processing: Concepts and Techniques|title-link=W:Transaction processing|date=1993|publisher}=Morgan Kaufmann|place=San Mateo, CA}} {{Refend}} 7zs0vn7kfperfzjt9ychfbp5qqnbd9r 2 246286 2691725 2627349 2024-12-13T01:14:28Z Dc.samizdat 2856930 /* 24-cell */ 2691725 wikitext text/x-wiki {{User sandbox}} == Polyscheme learning project == The Polyscheme learning project is intended to be series of articles on the regular polytopes and Euclidean spaces of 4 or more dimensions, that expand the corresponding series of Wikipedia encyclopedia articles to book length with textbook-like treatment of the material, additional learning resources and explanatory notes. == David Christie on Tolkien, to Ira Sandperl (emeritus) on Tolstoy == No doubt Tolkien's novels are not as profound as Tolstoy's with respect to our human condition, but Tolkien's address the more general condition of all peoples. He was the first novelist to write about the affairs of many peoples rather than just the comparatively parochial affairs of men. There is an expansive diversity to his interests, which shows his concern to transcend not just the patriarchy, but the humanitarchy. It disproves his critics' claims that his focus is not wide enough to include women, or people of color, or domains of experience (such as sex) which the critics feel are not sufficiently represented in his stories. == Regular convex 4-polytopes == {{W:Template:Regular convex 4-polytopes|wiki=W:}} {{Regular convex 4-polytopes|wiki=W:}} {| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes |- !align=right|[[W:Symmetry group|Symmetry group]] |align=center|[[W:Coxeter_group|A₄]] |align=center colspan=2|[[W:Hyperoctahedral_group|B₄]] |align=center|[[W:F4_(mathematics)|F₄]] |align=center colspan=2|[[W:H4_polytope|H₄]] |- !valign=top align=right|Name |valign=top align=center|[[W:5-cell|5-cell]]<BR> <BR> Hyper-<BR> [[W:Tetrahedron|tetrahedron]] |valign=top align=center|[[W:16-cell|16-cell]]<BR> <BR> Hyper-<BR> [[W:Octahedron|octahedron]] |valign=top align=center|[[W:8-cell|8-cell]]<BR> <BR> Hyper-<BR> [[W:Cube|cube]] |valign=top align=center|[[W:24-cell|24-cell]] |valign=top align=center|[[W:600-cell|600-cell]]<BR> <BR> Hyper-<BR> [[W:Icosahedron|icosahedron]] |valign=top align=center|[[W:120-cell|120-cell]]<BR> <BR> Hyper-<BR> [[W:Dodecahedron|dodecahedron]] |- !align=right|[[W:Schläfli symbol|Schläfli symbol]] |align=center|{3, 3, 3} |align=center|{3, 3, 4} |align=center|{4, 3, 3} |align=center|{3, 4, 3} |align=center|{3, 3, 5} |align=center|{5, 3, 3} |- !align=right|[[W:Coxeter-Dynkin diagram|Coxeter diagram]] |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} |- !valign=top align=right|Graph |align=center|[[Image:4-simplex t0.svg|120px]] |align=center|[[Image:4-cube t3.svg|121px]] |align=center|[[Image:4-cube t0.svg|120px]] |align=center|[[Image:24-cell t0 F4.svg|120px]] |align=center|[[Image:600-cell graph H4.svg|120px]] |align=center|[[Image:120-cell graph H4.svg|120px]] |- !align=right|Vertices |align=center|5 |align=center|8 |align=center|16 |align=center|24 |align=center|120 |align=center|600 |- !valign=top align=right|Edges |align=center|10 |align=center|24 |align=center|32 |align=center|96 |align=center|720 |align=center|1200 |- !valign=top align=right|Faces |align=center|10<BR>triangles |align=center|32<BR>triangles |align=center|24<BR>squares |align=center|96<BR>triangles |align=center|1200<BR>triangles |align=center|720<BR>pentagons |- !valign=top align=right|Cells |align=center|5<BR>tetrahedra |align=center|16<BR>tetrahedra |align=center|8<BR>cubes |align=center|24<BR>octahedra |align=center|600<BR>tetrahedra |align=center|120<BR>dodecahedra |- !valign=top align=right|[[W:Cartesian coordinates|Cartesian]]{{Efn|The coordinates (w, x, y, z) of the unit-radius origin-centered 4-polytope are given, in some cases as {permutations} or [even permutations] of the coordinate values.}}<BR>coordinates |align=center|{{font|size=85%|( 1, 0, 0, 0)<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})}} |align=center|{{font|size=85%|(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})<BR>([±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0])}} |align=center| |- !valign=top align=right|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf]]{{Efn|name=Hopf coordinates key|[[W:3-sphere#Hopf coordinates|Hopf spherical coordinates]]{{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.{{Sfn|Sadoc|2001|pp=575-576|loc=§2.2 The Hopf fibration of S3}} A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two completely orthogonal invariant planes{{Efn|The point itself (𝜉_''i'', 𝜂, 𝜉_''j'') does not necessarily lie in either of the invariant planes of rotation referenced to locate it (by convention, the ''wz'' and ''xy'' Cartesian planes), and never lies in both of them, since completely orthogonal planes do not intersect at any point except their common center. When 𝜂 {{=}} 0, the point lies in the 𝜉_''i'' "longitudinal" ''wz'' plane; when 𝜂 {{=}} {{sfrac|𝜋|2}} the point lies in the 𝜉_''j'' "equatorial" ''xy'' plane; and when 0 < 𝜂 < {{sfrac|𝜋|2}} the point does not lie in either invariant plane. Thus the 𝜉_''i'' and 𝜉_''j'' coordinates number vertices of two completely orthogonal great circle polygons which do ''not'' intersect (at the point or anywhere else).|name=reference planes of rotation}} which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} The 𝜉_''i'' and 𝜉_''j'' coordinates range over the vertices of completely orthogonal great circle polygons which do not intersect at any vertices. Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<BR> : w {{=}} cos 𝜉_''i'' sin 𝜂<BR> : x {{=}} cos 𝜉_''j'' cos 𝜂<BR> : y {{=}} sin 𝜉_''j'' cos 𝜂<BR> : z {{=}} sin 𝜉_''i'' sin 𝜂<BR> The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).<BR> The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], denoted SO(4), generates those polytopes. A rotation in 4D of a point {{math|{''ξ''_i, ''η'', ''ξ''_j<nowiki>}</nowiki>}} through angles {{math|''ξ''₁}} and {{math|''ξ''₂}} is simply expressed in Hopf coordinates as {{math|{''ξ''_i + ''ξ''₁, ''η'', ''ξ''_j + ''ξ''₂<nowiki>}</nowiki>}}.}} of the vertices are given as three independently permuted coordinates:<BR> :(𝜉_''i'', 𝜂, 𝜉_''j'')_𝑚<BR> where {<''k''} is the {permutation} of the ''k'' non-negative integers less than ''k'', and {≤''k''} is the permutation of the ''k''+1 non-negative integers less than or equal to ''k''. Each coordinate permutes one set of the 4-polytope's great circle polygons, so the permuted coordinate set expresses one set of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-space]] which generates the 4-polytope. With Cartesian coordinates the choice of radius is a parameter determining the reference frame, but Hopf coordinates are radius-independent: all Hopf coordinates convert to unit-radius Cartesian coordinates by the same mapping. {{Efn|name=Hopf coordinates conversion}} Unlike Cartesian coordinates, Hopf coordinates are not necessarily unique to each point; there may be Hopf coordinate synonyms for a vertex. The multiplicity 𝑚 of the coordinate permutation is the ratio of the number of Hopf coordinates to the number of vertices.}}<BR>coordinates |align=center|[[User:Dc.samizdat/sandbox#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |align=center|{{font|color=green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}</small>)₁}} |- !valign=top align=right|Long radius{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope.|name=polytopes ordered by size and complexity}} |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |- !valign=top align=right|Edge length |align=center|<small>{{sfrac|{{radic|5}}|{{radic|2}}}}</small> ≈ 1.581 |align=center|{{radic|2}} ≈ 1.414 |align=center|1 |align=center|1 |align=center|<small>{{sfrac|1|ϕ}}</small> ≈ 0.618 |align=center|<small>{{Sfrac|1|{{radic|2}}ϕ²}}</small> ≈ 0.270 |- !valign=top align=right|Short radius |align=center|{{sfrac|1|4}} |align=center|{{sfrac|1|2}} |align=center|{{sfrac|1|2}} |align=center|<small>{{sfrac|{{radic|2}}|2}}</small> ≈ 0.707 |align=center|<small>1 - ({{sfrac|{{radic|2}}|2{{radic|3}}φ}})²</small> ≈ 0.936 |align=center|<small>1 - ({{sfrac|1|2{{radic|3}}φ}})²</small> ≈ 0.968 |- !valign=top align=right|Area |align=center|<small>10•{{sfrac|{{radic|8}}|3}}</small> ≈ 9.428 |align=center|<small>32•{{sfrac|{{radic|3}}|4}}</small> ≈ 13.856 |align=center|24 |align=center|<small>96•{{sfrac|{{radic|3}}|8}}</small> ≈ 20.785 |align=center|<small>1200•{{sfrac|{{radic|3}}|8φ²}}</small> ≈ 99.238 |align=center|<small>720•{{sfrac|25+10{{radic|5}}|8φ⁴}}</small> ≈ 621.9 |- !valign=top align=right|Volume |align=center|<small>5•{{sfrac|5{{radic|5}}|24}}</small> ≈ 2.329 |align=center|<small>16•{{sfrac|1|3}}</small> ≈ 5.333 |align=center|8 |align=center|<small>24•{{sfrac|{{radic|2}}|3}}</small> ≈ 11.314 |align=center|<small>600•{{sfrac|1|3{{radic|8}}φ³}}</small> ≈ 16.693 |align=center|<small>120•{{sfrac|2 + φ|2{{radic|8}}φ³}}</small> ≈ 18.118 |- !valign=top align=right|4-Content |align=center|<small>{{sfrac|{{radic|5}}|24}}•({{sfrac|{{radic|5}}|2}})⁴</small> ≈ 0.146 |align=center|<small>{{sfrac|2|3}}</small> ≈ 0.667 |align=center|1 |align=center|2 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 3.907 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 4.385 |} == Scratch == 400 <math>\sqrt{5} \curlywedge (2-\phi)</math> hexagons In each hemi-icosahedron, 15 edges come from 10 disjoint 5-cells each contributing 4 edges to this hemi-icosahedron, and 3 hemi-icosahedra fitting together around each edge sharing it, as Grünbaum discovered they do. One other hemi-icosahedron fits against each of 10 hemi-icosahedron faces, and two other hemi-icosahedra fit around each of 15 opposite edge, all the same set of 11 hemi-icosahedra. == Coordinate systems on the 3-sphere == The four Euclidean coordinates for {{math|''S''³}} are redundant since they are subject to the condition that {{math|1=''x''₀² + ''x''₁² + ''x''₂² + ''x''₃² = 1}}. As a 3-dimensional manifold one should be able to parameterize {{math|''S''³}} by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as [[latitude]] and [[longitude]]). Due to the nontrivial topology of {{math|''S''³}} it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use ''at least'' two [[coordinate chart]]s. === Hopf coordinates of the regular convex 4-polytopes === As with Cartesian coordinates, there are multiple reference frames which give useful Hopf coordinates.{{Efn|name=Hopf coordinates}} One can choose any of the 4-polytope's great circle polygons for the 𝜉_''i'' coordinate to range over; then the 𝜉_''j'' coordinate will range over the vertices of whatever kind of great circle polygon lies orthogonal to the 𝜉_''i'' great circle plane. Note that the 𝜉_''j'' polygon will sometimes be a digon (a great circle plane intersecting only 2 vertices), as in the case of the the planes orthogonal to the 24-cell's hexagonal planes. The choice of polygons will (almost) determine the only possible range for the 𝜂 coordinate; the only remaining variable is the multiplicity 𝓂 of the coordinates. {| class="wikitable mw-collapsible {{{collapsestate|mw-expanded}}}" style="white-space:nowrap" !colspan=8|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Efn|name=Hopf coordinates key}} of the regular convex 4-polytopes |- ! ![[W:5-cell|5-cell]] ![[W:16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[W:24-cell|24-cell]] ![[W:600-cell|600-cell]] ![[120-cell]] |- !Digons |[[#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] | | | | |{{font color|green|(<small>{<2}𝜋, {<150}{{sfrac|𝜋|300}}, {<2}𝜋</small>)₁}} |- !Squares | |[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] | |rowspan=2|[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]]<BR>[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<15}{{sfrac|𝜋|30}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<75}{{sfrac|𝜋|150}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |- !Rectangles | | |[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] | | !Pentagons | | | | |[[#Great circle pentagons of the 600-cell|{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₅}}]] |{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₁}} |- !Hexagons | | | |[[#Great circle hexagons and squares of the 24-cell|{{font color|red|({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<5}{{sfrac|𝜋|10}}, {<4}{{sfrac|𝜋|2}}</small>)₁}} |[[#Great circle squares and hexagons of the 120-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤24}{{sfrac|𝜋|48}}, {<4}{{sfrac|𝜋|2}}</small>)₁}}]] |- ! | | | |[[#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<10}{{sfrac|𝜋|20}}, ({<6}{{sfrac|𝜋|3}}</small>)₃}} |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}},{<50}{{sfrac|𝜋|100}},{<6}{{sfrac|𝜋|3}}</small>)₃}} |- !Decagons | | | | |[[#Great circle decagons and hexagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤1}{{sfrac|𝜋|2}}, {<6}{{sfrac|𝜋|3}}</small>)₁}}]] |{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {<10}{{sfrac|𝜋|20}}, {<6}{{sfrac|𝜋|3}}</small>)₁}} | |- ! | | | | |[[#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |[[#Great circle decagons of the 120-cell|{{font color|red|(<small>{<10}{{sfrac|𝜋|5}},{≤5}{{sfrac|𝜋|10}},{<10}{{sfrac|𝜋|5}}</small>)₁}}]] |- !30-gons | | | | | |{{font color|green|(<small>{<30}{{sfrac|𝜋|15}},{≤15}{{sfrac|𝜋|30}},{<30}{{sfrac|𝜋|15}}</small>)₂₄}} |} ==== Great circle digons of the 5-cell==== One set of Cartesian origin-centered [[W:5-cell#Construction|coordinates for the 5-cell]] can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}} and radius {{radic|3.2}}: ( {{sfrac|4|{{radic|5}}}}, 0, 0, 0) (−{{sfrac|1|{{radic|5}}}}, 1, 1, 1) (−{{sfrac|1|{{radic|5}}}}, 1,−1,−1) (−{{sfrac|1|{{radic|5}}}},−1, 1,−1) (−{{sfrac|1|{{radic|5}}}},−1,−1, 1) Rescaled to unit radius and edge length {{sfrac|{{radic|5}}|{{radic|2}}}} these coordinates are: ( 1, 0, 0, 0) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) {| class="wikitable" !colspan=2|Great circle digons of the 5-cell<BR> Cartesian{{s|2}}({ , , , })<BR> Hopf{{s|2}}(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( 1, 0, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 1{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |(-{{sfrac|1|4}},-{{sfrac|{{radic|5}}|4}}, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|4}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|5}}|4}} ≈ 0.559}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 3{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 4{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} | |} ==== Great squares of the 16-cell ==== {| class="wikitable floatright" | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''xy'' plane |- |( 0, 1, 0, 0)||( 0, 0,-1, 0) |- |( 0, 0, 1, 0)||( 0,-1, 0, 0) |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''wz'' plane |- |( 1, 0, 0, 0)||( 0, 0, 0,-1) |- |( 0, 0, 0, 1)||(-1, 0, 0, 0) |} |}The 8 vertices of the 16-cell lie on the 4 coordinate axes and form 6 great squares in the 6 orthogonal central planes. The Cartesian axes lie on the diagonals of the square tables, which resemble the great squares. Rotate the tables 45 degrees clockwise for a vertex up orientation, and another 90 degrees for the standard ''xy'' orientation. By convention rotations are always specified in two completely orthogonal invariant planes xy (whose vertices are numbered counterclockwise by 𝜉_''xy'') and wz (whose vertices are numbered counterclockwise by 𝜉_''wz''). The rotation in the xy plane does not move points in the wz plane, and vice versa. In the 16-cell these two simple rotations rotate disjoint sets of 4 vertices each (because completely orthogonal planes intersect only at the origin and share no vertices). The 𝜂 coordinate of the 4 vertices in the xy plane is 0 and the 𝜂 coordinate of the 4 vertices in the wz plane is 1 ({{sfrac|𝜋|2}}). The w and z coordinates of the vertices in the xy plane are 0 regardless of the rotational position of the wz plane (the 𝜉_''wz'' coordinate), and the x and y coordinates of the vertices in the wz plane are 0 regardless of the rotational position of the xy plane (the 𝜉_''xy'' coordinate). {| class="wikitable" !colspan=2|Great squares of the 16-cell<BR> Cartesian{{s|2}}({0, ±1, 0, 0})<BR> Hopf{{s|2}}(<small>{<3}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<3}{{sfrac|𝜋|2}}</small>) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''xy'' plane |- !(𝜉_''xy'', 0, 0)||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 0, 1, 0, 0)||( 0, 0,-1, 0) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 1, 0)||( 0,-1, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}0: 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}0: 1}} |} | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''wz'' plane |- !(0, {{sfrac|𝜋|2}}, 𝜉_''wz'')||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 1, 0, 0, 0)||( 0, 0, 0,-1) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 0, 1)||(-1, 0, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 0}} |} |} ====Great rectangles (60 degree planes) of the 8-cell==== None of the 8-cell's 16 vertices lie in the 6 orthogonal central planes. The "north pole" is not a vertex, and 0 does not appear as a value in any (Hopf or Cartesian) coordinate. Each of the 8-cell's eight {{radic|4}} long diameters joining two antipodal vertices lies 45 degrees ({{sfrac|𝜋|4}}) off each of the 4 Cartesian coordinate axes. The Hopf 𝜼 coordinate is {{sfrac|𝜋|4}} for all the vertices, and the 𝜉_''xy'' and 𝜉_''wz'' coordinates are even and odd multiples of {{sfrac|𝜋|4}} respectively. Although the 16 vertices do not lie in the 6 orthogonal central planes, they do lie (by fours) in central planes, but the central polygons they form are rectangles (not squares), and the planes are inclined at 60 degrees ({{sfrac|𝜋|3}}) to each other and to the orthogonal central planes. These 16 great rectangles measure {{radic|1}} by {{radic|3}}, and their {{radic|1}} edges are opposite pairs of 8-cell edges. Their {{radic|3}} edges are interior chords of the 8-cell: long diagonals of the 8 cubic cells.{{Efn|In the 24-cell we find these 16 central rectangles inscribed in the 16 central hexagons. The 8-cell's great rectangles are the same central planes as the 24-cell's great hexagons, but the 8-cell has only 4 of the 6 hexagon vertices. The 16-point (8-cell) is a 24-point (24-cell) with 8 points (square pyramids) lopped off.}} Because there is only one 𝜼 coordinate value, only one table is required, but note that the table is not a great square but a duocylinder (its opposite edges are identified).{{Efn|For fixed {{mvar|η}} Hopf coordinates describe a torus parameterized by {{math|''ξ''_''xy''}} and {{math|''ξ''_''wz''}}, with {{math|''η'' {{=}} {{sfrac|π|4}}}} being the special case of the [[W:Clifford torus|Clifford torus]] in the {{mvar|xy}}- and {{mvar|wz}}-planes. All vertices of the 8-cell lie on the Clifford torus, a "flat" 2-dimensional surface embedded in the 3-sphere. The Clifford torus divides the 3-sphere into two congruent ''solid'' tori. [[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|In a stereographic projection]], the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized.}} Each row, column or diagonal in the table is a great rectangle. {| class="wikitable" ! colspan="2" |Great rectangles (60 degree planes) of the 8-cell Cartesian{{s|2}}(<small>±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}</small>)<BR> Hopf{{s|2}}({0 2 4 6}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}) |- | {| class="wikitable" !(𝜉_''xy'', {{sfrac|𝜋|4}}, 𝜉_''wz'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !0{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} |- !2{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !4{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%| 0}} !{{font|size=75%|−1}} |- !6{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%|−1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} ==== Great squares and rectangles of the 24-cell ==== The great square coordinates of the 16-cell (above), combined with the great rectangle coordinates of the 8-cell (above), comprise a set of coordinates for the 24-cell. Because the 16-cell lies vertex-up in this coordinate system, so does the 24-cell. ==== Great squares of the 24-cell ==== Another useful set of coordinates for the 24-cell is comprised solely of orthogonal great squares. In this coordinate system the 24-cell lies cell-up, and the great squares are aligned with the squares of the coordinate lattice. {| class="wikitable" !Great squares of the 24-cell<BR>Cartesian{{s|2}}(<small>±{{radic|{{sfrac|1|2}}}}, ±{{radic|{{sfrac|1|2}}}}, 0, 0</small>) |- | {| class="wikitable" !Hopf{{s|2}}({<4}{{sfrac|𝜋|2}}, {{sfrac|𝜋|4}}, {<4}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" !(𝜉_''i'', {{sfrac|𝜋|4}}, 𝜉_''j'') !0{{sfrac|𝜋|2}} !1{{sfrac|𝜋|2}} !2{{sfrac|𝜋|2}} !3{{sfrac|𝜋|2}} !sin !cos |- !0{{sfrac|𝜋|2}} |(<small> {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%| 1}} |- !1{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !2{{sfrac|𝜋|2}} |(<small>-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}},-0, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}},-0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%|-1}} |- !3{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} |- | {| class="wikitable" !Hopf{{s|2}}({1 3 5 7}{{sfrac|𝜋|4}}, {≤1}{{sfrac|𝜋|2}}, {1 3 5 7}{{sfrac|𝜋|4}})₄ |- | {| class="wikitable" !(𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" !(𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} |} |} ==== Great squares and rectangles of the compound of dual 24-cells ==== Two sets of coordinates for the 24-cell have now been given (above). In the first (great squares and rectangles) the 24-cell lies vertex-up, and in the second (great squares only) it lies cell-up. The 24-cell being a self-dual 4-polytope, these two 24-cells are duals of each other, the vertices of one lying at the cell centers of the other, and the union of their two sets of coordinates is a 48-vertex compound of duals of the same radius. In this compound of two 24-cells, 24 16-cells are inscribed: the 3 inscribed in each of the dual 24-cells, and 18 others which span the two 24-cells.{{Sfn|Waegell|Aravind|2009}} Each of these 24 16-cells, with its 4 orthogonal axes and 6 orthogonal planes, constitutes an equivalent ''basis'' for a Cartesian coordinate system, and contains three pairs of completely orthogonal planes, each pair of which intersects all 8 of the 16-cell's vertices. The compound of two 24-cells has 24 axes and 24 bases, with each basis consisting of four axes and each axis occurring in four bases. The 3 16-cells inscribed in each 24-cell are disjoint from each other and the dual 24-cell's 3 inscribed 16-cells. Each 24-cell has 18 central squares, and the 18 spanning 16-cells are each comprised of 4 vertices comprising a great square from one of the 24-cells, and another 4 vertices comprising a great square from the dual 24-cell.{{Efn|In each of the 18 16-cells, the two central squares from dual 24-cells are orthogonal, but not completely orthogonal. Each is already completely orthogonal to another central square within the same 24-cell (within the same 16-cell), and in 4-space a plane cannot be completely orthogonal to more than one other plane through the same point. Although their two sets of 4 vertices are disjoint, that is not because their square planes are completely orthogonal; rather, their planes intersect in a line, but their vertices remain disjoint because the line of intersection does not pass through any of their vertices.}} ==== Great hexagons and squares of the compound of dual 24-cells ==== In a single 24-cell, the hexagonal central planes lie at 60 degrees to each other and to the square central planes. The central planes orthogonal to the hexagonal planes are digons: they intersect only 2 vertices. Consequently it is not possible to find Hopf coordinates for the 24-cell in which both the 𝜉_''i'' and 𝜉_''j'' orthogonal invariant planes contain hexagons. However, the 24-cell and its unscaled dual form a compound in which the dual 24-cells are separated by a Clifford displacement (an isoclinic rotation) of 30 degrees. The hexagonal planes are still not orthogonal to each other in this compound (they are inclined at 30 degrees or 60 degrees to each other), so it is still impossible to find a 6 x 6 array of Hopf coordinates, but the hexagonal planes of one 24-cell are orthogonal to the square planes of the other. In this compound of 48 vertices, a hexagonal wz plane and a square xy plane can be the invariant planes of a 6 x 4 array of Hopf coordinates. {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the compound of dual 24-cells Cartesian{{s|3}}(<small>±_, ±_, _, _</small>)<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {1 2}{{sfrac|𝜋|6}}, {<small><4</small>}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" style="white-space:nowrap;" !|(𝜉_''i'', 1{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|2}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|1|2}}}} |} |} ==== Great hexagons and digons of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and digons of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜋, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} -1}} |} |} ==== Great hexagons and squares of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 0𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( 0, 1, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |( 0, {{sfrac|1|2}}, 0, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{sfrac|3|4}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{sfrac|3|4}}, 0) |( 0, 0, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( 1, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |( {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 1𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}|| {{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( a, , 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} -1}} |} |} ==== Dual fibrations ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[fiber bundle]] of Clifford parallel great circles is a discrete [[Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once. {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell<BR> Hopf ({<<small>6</small>}{{sfrac|𝜋|3}}, {<small><10</small>}{{sfrac|𝜋|20}}, {<small><2</small>}𝜋)₁<BR> Cartesian ({<small>0, ±1, 0, 0</small>}) (<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>) ([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle pentagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle pentagons of the 600-cell<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>])<BR> Hopf{{s|3}}({<small>0 2 4 6 8</small>}{{sfrac|𝜋|5}}, {<small><24</small>}{{sfrac|𝜋|48}}), {<small>1 3 5 7 9</small>}{{sfrac|𝜋|5}})₅ |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'',{{sfrac|𝜋|4}}, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle squares and hexagons of the 120-cell==== {| class="wikitable" !colspan=1|Great circles of the 120-cell:<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {<small>≤24</small>}{{sfrac|𝜋|48}}, {<small><4</small>}{{sfrac|𝜋|2}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]){{s|3}}... |- | |} ====Great circle decagons and hexagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell:<BR> Hopf{{s|3}}({<10}{{sfrac|𝜋|5}}, {<small>≤1</small>}{{sfrac|𝜋|2}}, {<<small>6</small>}{{sfrac|𝜋|3}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle decagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons of the 600-cell:<BR> Hopf{{s|3}}({<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}}, {<small>0 1 2 3 4 5</small>}{{sfrac|𝜋|10}}, {<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}})₅<BR> Cartesian{{s|3}}... |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 0{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0))||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !style="white-space:nowrap;"|{{font|size=75%|sin}}||{{font|size=75%|𝜂 {{=}} 0}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !style="white-space:nowrap;"|{{font|size=75%|cos}}||{{font|size=75%|𝜂 {{=}} 1}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 1{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}},{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, 0)||(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}},-{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}},-{{sfrac|1|4}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ^-1|2}} ≈ 0.309}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} b ≈ 0.951}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} {{s|5}}<small>{{sfrac|bϕ|2}} ≈ 0.769{{s|5}}ab ≈ 0.559{{s|5}}{{sfrac|bϕ^-1|2}} ≈ 0.294{{s|5}}b² ≈ 0.905</small> |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 2{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 3{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 4{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 5{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |} == Equilateral rings == Equilateral rings are those which can be constructed out of equilateral triangles on the circumference of a sphere. === Borromean equilateral rings === The {1,1,1} torus knot. The vertices of the regular icosahedron form five sets of three concentric, mutually [[orthogonal]] [[golden rectangle]]s, whose edges form [[Borromean rings]]. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures <math>2\times 4</math>. These three rectangles are the shortest possible representation of the Borromean rings using only edges of the [[integer lattice]]. ... == Kinematics == In 3D we have the kinematic transformations of the cuboctahedron (cuboctahedron, icosahedron, jessen's, golden icosa?, octahedron-2, tetrahedron-4?) and their duals, the transformations of the dodecahedron: two sets of nesting Russian dolls (or perhaps one set?). In 4D we apparently have instances of the cuboctahedron nestings in the 600-cell (and perhaps the dodecahedron nestings as well, in the 120-cell?). This suggests that the unit-radius sequence of 4-polytopes may contain dynamic as well as static nestings. From [[W:Kinematics of the cuboctahedron#Duality of the rigid-edge and elastic-edge transformations|Kinematics of the cuboctahedron § Duality of the rigid-edge and elastic-edge transformations]]: <blockquote> Finally, both transformations are pure abstractions, the two limit cases of an infinite family of cuboctahedron transformations in which there are two elasticity parameters and no requirement that one of them be 0. ... In engineering practice, only a tiny amount of elasticity is required to allow a significant degree of motion, so most tensegrity structures are constructed to be "drum-tight" using nearly inelastic struts ''and'' cables. A '''tensegrity icosahedron transformation''' is a kinematic cuboctahedron transformation with reciprocal small elasticity parameters.</blockquote> From [[W:24-cell#Double rotations|24-cell § Double rotations]]: <blockquote>The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes of rotation at once.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. </blockquote> This suggests that the reciprocal limit-case kinematic transformations (with one orthogonal elasticity parameter or the other equal to 0) may be expressable as double rotations, considering the relativity of such transformations.{{Efn|name=transformations}} === Completely orthogonal planes === In three dimensions (on polyhedra) there are no disjoint great circles. Every pair of great circles intersects at two points, the endpoints of a diameter of the sphere. But in four dimensions (on polychora) every great circle is disjoint from exactly one other great circle: the one to which it is completely orthogonal. If we consider the two polyhedral great circles' common diameter to be a common axis of rotation, we can see that rotating either circle about that axis generates the whole polyhedron; thus either circle by itself can generate the whole polyhedron by rotation. But in four dimensions two completely orthogonal great circles have no common axis of rotation (no points at all in common, all their points are disjoint). Clearly either circle by itself cannot generate the entire polychoron by rotation about a fixed axis. Rotating each circle about an axis generates only half the points on the 3-sphere - rotating the other circle generates the other half of them. Rotation about a fixed axis in four dimensions necessarily leaves an entire plane fixed, and generates only a 3-dimensional polyhedron. If the great circle in the xy plane is rotated about the y axis, only a 2-sphere is generated, and all the points on the 3-sphere outside the hyperplane w = 0 will be left out. In the 24-cell and the 8-cell, which are radially equilateral, ... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. {{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Coxeter mirrors ==== [[W:Coxeter group|Coxeter group]] theory (the mathematics of [[W:Polytopes|polytopes]] of any number of dimensions) can be informally described as a ''finite closed system of mirrors'', or the [[W:Geometry|geometry]] of multiple [[W:Mirror#Mirror images|mirror images]]. Mathematically it is the equivalent of the theory of finite [[W:Reflection group|reflection groups]] and [[W:Root system|root systems]], expressed in a different mathematical language. But unlike those [[W:Group theory|group theory]] languages, its principal objects can be defined in the most intuitive and elementary way: <blockquote> Imagine a few (semi-transparent) mirrors in ordinary three dimensional space. Mirrors (more precisely, their images) multiply by reflecting in each other, like in a [[W:Kaleidoscope|kaleidoscope]] or a [[W:Gallery of mirrors|gallery of mirrors]]. A ''closed system of mirrors'' is what we see when we look into such a kaleidoscope.{{Sfn|Borovik|2006|loc=§1. Mirrors and Reflections|pp=18-19}}</blockquote> Coxeter refers to the space between two parallel mirrors as "the region of possible objects". This suggests that space itself is being generated by the objects that it contains, which parallel reflections multiply infinitely. When the mirrors are not parallel, the multiplication may be finite rather than infinite (provided the dihedral angle between the mirrors is a submultiple of 𝝅). The space between two or more such ''intersecting'' mirrors is called the fundamental region, and it constitutes a proto-space of a finite number of dimensions with mirrors as its bounding walls. Coxeter says "the lines of symmetry or circles of symmetry or planes of symmetry are mirrors reflecting the whole pattern into itself. We count these circles of symmetry by counting their pairs of antipodal points of intersection with a single equator."{{Sfn|Coxeter|1938}} He characterizes great circles of symmetry in terms of the Petrie ''h''-gon i.e. the 3''h''/2 circles of symmetry possessed by each Platonic solid. For example the Petrie polygon of the octahedron and the cube is the hexagon, so they have 18/2 = 9 great circle planes of symmetry (mirrors). Each is generated by placing a single point-object in the fundamental region (off the surface of any mirrors); reflected in the mirrors it multiplies into all the vertices of the polytope. ==== Translation-rotations ==== An object displacement in space may be a rotation (which leaves at least one point invariant), a translation (which does not), or a combination of both.{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations|pp=33-38}} The circular path of a rotation may be combined with a translation in the axial direction of the rotation yielding a ''screw-displacement'', the general case of a displacement.{{Sfn|Coxeter|1973|loc=§3.14|ps=; "''Every displacement is a screw-displacement'' (including, in particular, a rotation or a translation)."|p=38}} In three dimensions, a screw-displacement is a simple helix, as its name suggests. In four dimensions, the circular path of an isoclinic rotation is already a helix (a geodesic isocline), and there are four orthogonal axial directions of rotation.{{Sfn|Coxeter|1973|loc=§12.1 Orthogonal transformations|pp=213-217|ps=; "The general displacement preserving the origin in four dimensions is a ''double rotation''.... The two completely orthogonal planes of rotation are uniquely determined except when 𝜉₂ {{=}} 𝜉₁, in which case... we have a ''Clifford displacement''."}} In the unit-radius 24-cell an isoclinic rotation by 60° moves each vertex {{radic|3/4}} ≈ 0.866 in each of four orthogonal directions at once, a total Pythagorean distance as if it had moved straight along a combined {{radic|3}} ≈ 1.732 chord. When the rotation is combined with a unit translation, the {{radic|1}} translation vector must be divided among all four rotation vectors. The vertex moves {{radic|3/4 + 1/4}} = {{radic|1}} in each of the four orthogonal directions, moving a combined Pythagorean distance of {{radic|4}}, the maximum ''unit displacement'' in 4 dimensions: the distance which is the [[W:Tesseract#Radial equilateral symmetry|long diameter of the 4-hypercube (tesseract)]]. This movement ''could'' take the vertex to its antipodal vertex {{radic|4}} away, if the direction of the translation is so aligned, but in all other cases it will take it to a point outside the 24-cell. ==== Relative screw displacement ==== (screw displacement) QT = QR² = R⁴ = Q² (double rotation){{Sfn|Coxeter|1973|P=217|loc=§12.2 Congruent transformations}} A screw displacement in four dimensions is equivalent to a double rotation, by the principle of relativity. There are only two kinds of screw displacements possible in only ''four'' dimensions: the single kind (which also occurs in three dimensions), and the double kind (which requires four dimensions). The latter kind of screw displacement is inherently double, the product of 4 reflections, just like a double rotation. In fact it ''is'' just a double rotation, seen from a moving inertial reference frame. A product of two reflections is a (simple) rotation, unless the reflecting facets are exactly parallel, in which case it is a translation. In other words, a translation is just a rotation on a circle of infinitely long radius (a straight line). A screw displacement is just a double rotation in which one rotation (the one which is the translation) has infinitely long radius, i.e. a vanishingly small angle of rotation (near 0 degrees) compared to the angle of the other rotation (between 0 and 90 degrees). The screw displacement looks like a simple rotation within a three dimensional reference frame that is in uniform translation on the 4th dimension axis at near-infinite velocity. In the case of actual moving objects, no actual translation is an infinite straight line, and no velocity is infinite; any moving object that describes a screw-displacement is presumably moving on a curved translation under the influence, at least, of some distant gravitational force, however miniscule, and the radius of the translation-rotation it describes is merely very long, not infinite. So we can say that there can be only one range of situations in actuality, no perfectly straight translations but only double rotations of more or less eccentricity, which will appear to be simple rotations inside a 3-dimensional reference frame moving uniformly along the 4th dimension "translation" axis. Its eccentricity is merely a matter of choice of reference frame: it looks like a simple rotation in a reference frame moving uniformly with the translation, and a double rotation (of perhaps extreme eccentricity) in a reference frame that is not moving with the translation. Within this range of possibilities, only one possibility is ''not'' eccentric: the case of the equi-angled double rotation, called an isoclinic rotation or Clifford displacement. Since the ratio of eccentricity is a matter of choice of reference frame, we may adopt as our preferred reference frame (of any actual screw displacement occuring in practice) the inertial reference frame in which the double rotation is isoclinic. ==== Total internal reflection ==== The phenomenon in physics known as [[W:Total internal reflection|total internal reflection]] keeps light confined within one strand of a [[W:Optical fiber|fiber optic cable]]. Isoclinic rotations and screw displacements in 4-dimensional space are both the consequence of four symmetrical reflections, and their propagation corresponds to a total internal reflection within the 3-sphere. Consequently 4-dimensional space itself acts as a [[W:waveguide|waveguide]] for isoclinically rotating objects during their translation. This provides a purely geometric model for the [[w:inertia|inertia]] of mass-carrying objects, and for light-wave propagation. ==== Isoclines ==== In an isoclinic rotation the vertices of a 4-polytope such as the 24-cell move on ''isoclines'', which are helical circles that wind through all four dimensions. Isoclinic rotations are [[W:chiral|chiral]], occuring in left-handed and right-handed mirror-image pairs in which the moving vertices reach different destinations along left or right paths. The isoclines themselves however (the helical paths of the moving vertices) are not chiral objects: they are non-twisted (directly congruent) ''circles'', of a special 4-dimensional kind. Every left isocline path in a left-handed rotation acts also as a right isocline path in some right-handed rotation, in some ''other'' left-right pair of isoclinic rotations. Isoclinic rotations and their isoclines occur as fibrations (fiber bundles of non-intersecting but interlinked circles), with each fibration consisting of a single distinct left-right pair of isoclinic rotations. Each distinct left (or right) rotation has some number of isoclines, which are the circular paths along which its vertices orbit, with each vertex confined to a single isocline circle throughout the rotation. The multiple isoclines of a distinct left or right rotation do not intersect each other; they are Clifford parallel, which means that they are curved lines which are parallel to each other, in the sense that they are the same distance apart at all of their corresponding (nearest) points. Thus the moving vertices in an isoclinic rotation circulate in parallel disjoint sets. In the 24-cell's characteristic kind of isoclinic rotation, the moving vertices circulate on skew hexagon isoclines, in 4 parallel disjoint sets of 6 moving vertices each. This characteristic kind of isoclinic rotation occurs in four different fibrations: there are four distinct left-right rotation pairs. In each distinct left (or right) rotation, there are 4 Clifford parallel isoclines, each of which is a helical circle through 6 vertex positions. The 4 disjoint circles of 6 vertices pass through all 24 vertices of the 24-cell, just once. Although the four isocline circles do not intersect, they do pass through each other as do the links of a chain, but unlike linked circles in three-dimensional space, they all share the same center point. ==== Polygrams and cell rings ==== The isoclines of [[W:24-cell#Isoclinic rotations|24-cell isoclinic rotations]] in ''hexagonal'' central planes have 6 chords which form a [[W:Skew polygon|skew]] [[W:hexagram|hexagram]]. Every [[W:24-cell#Helical hexagrams and their isoclines|hexagram isocline]] is contained within the volume of a distinct [[W:24-cell#Cell rings|ring of 6 face-bonded octahedral cells]] which, like its axial great circle hexa''gon'' is an equatorial [[W:24-cell#Rings|ring of the 24-cell]]. Each 6-cell ring contains the left and right isoclines of a distinct left-right pair of isoclinic rotations. [[W:24-cell#6-cell rings|The 6-cell ring itself]] is not a chiral object because it contains ''both'' mirror-image isoclines: they are twisted in opposite directions (around each other), but the 6-cell ring that contains them both has no [[W:Torsion of a curve|torsion]]. ==== Reflections ==== Because each octahedral cell volume can be subdivided into 48 orthoschemes (the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the octahedron]] and [[W:24-cell#Characteristic orthoscheme|of the 24-cell]]), we can be more precise in describing the cell ring each isocline stays within. Within the 6-cell ring of face-bonded spherical octahedra is a ring of face-bonded spherical characteristic tetrahedra that contains the isocline. Several characteristics of this ring are evident from the nature of [[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoschemes]]: it consists of alternating left-hand and right-hand orthoschemes (mirror images of each other in their common face), and each of them contains one and only one vertex of the 24-cell.{{Efn|In the 24-cell 48 [[W:24-cell#Characteristic orthoscheme|characteristic 4-orthoschemes (5-cells)]] meet at each of the 24 vertices, and 48 [[W:Octahedron#Characteristic orthoscheme|characteristic 3-orthoschemes (tetrahedra)]] fill each of the 24 octahedral cells. The number of characteristic orthoschemes in a regular polytope is the ''order'' of its [[W:Coxeter group|symmetry group]], which for the octahedron is 48, and for the 24-cell is 1152 {{=}} 24 * 48.}} The sequence of orthoscheme vertices is the same as [[W:24-cell#Helical hexagrams and their isoclines|the sequence of isocline vertices]], except that it includes the vertices near-missed by the isocline as well as those the isocline intersects (12 distinct vertices instead of 6), and the orthoscheme sequence will have many more than 12 elements because it includes more than two orthoschemes incident to each vertex.{{Efn|The ring contains more orthoschemes than necessary to contain the isocline, because of our stipulation that all pairs of adjacent orthoschemes be face-bonded. The isocline intersects only a vertex, or only an edge, of some orthoschemes in the face-bonded ring. At each vertex it hits, the isocline passes between two orthoschemes that touch only at that vertex. Near each vertex that it misses, the isocline passes through an edge between two orthoschemes that touch only at that edge.}} === Physical space === We attempt to be more precise about the shape of this 4-space, and in particular, the cause of its shape, i.e. the relationship between the fundamental forces observed in nature and this spatial geometry. As Einstein did in his 1923 thought experiment, we identify the observed 3-dimensional cosmos (everything in it up to some large scale such as a galaxy) as a thin manifold embedded in a Euclidean (i.e. flat) 4-dimensional space of the kind elucidated by Coxeter. Further we postulate that every mass-carrying particle in this space is in motion at speed <math>c</math> relative to the 4-dimensional space itself. The 4-space therefore has a quasi-ether-like existence insofar as it defines a field at absolute rest, relative to which the motion of all particles at speed <math>c</math> can be universally compared, with the important provision that no particle, anywhere, is ever at absolute rest itself with respect to this field. The condition of absolute rest is an abstract condition attributable only to the field, and never to any tangible object. Thus the field itself (4-dimensional Euclidean space) is an abstraction somewhat more tangible than Mach's relative space, but much less tangible than the luminous ether, much as Einstein found 4-dimensional spacetime to be. Directions and distances can be fixed universally within the Euclidean 4-space field (they are invariant for all observers regardless of their direction of motion within the field), but locations can only be relative to some object (not to the field itself), and all 4-dimensional velocities are invariant: they are always <math>c</math> with respect to the field, for any mass-bearing particle or observer. Einstein's general relativity identifies gravity as a fictitious force, attributable to the shape of the 3-dimensional manifold rather than to an attractive force acting instantaneously at a distance. The 3-dimensional manifold is said to be singular and universal (all objects in the universe lie within it), but its shape varies by location. It is assumed to curve or dimple in the vicinity of massive objects, such that other objects fall into the dimples naturally in the course of following straight-line paths (geodesics) through it. In general relativity 3-dimensional space is flat near each observer, but there is no universally flat space except in regions far removed from massive objects, i.e. in places where the simplifications of the theory of special relativity can be assumed. But in Euclidean relativity this 3-dimensional manifold is embedded in a 4-dimensional Euclidean space, and that 4-space field is flat universally, at all times for all observers. Furthermore, we only assume that the 4-dimensional space is singular and universal; there might be more than one 3-dimensional manifold embedded within it, and the 3-manifolds do not necessarily intersect. In Euclidean relativity we expect that not just gravity, but all the fundamental forces observed in nature, are an expression of the local geometry of a 3-space manifold embedded in Euclidean 4-space. By ''expression'' we mean the consequence of a transformation such as a reflection, rotation or translation, i.e. operations of the fundamental Coxeter symmetry groups, which characterize the Euclidean geometry of the universal space in which the 3-space manifolds are embedded. === Closed 3-manifolds embedded in 4-space === The only reason to suppose there is only one such closed, curved soap-bubble 3-manifold in our 4-space universe is the assumption that every particle in the universe had a common origin, at a single point in 4-space and a single moment in time in a big bang, and even in that case there could be many such soap-bubble 3-manifolds in existence now. One can certainly model the observed universe as a single closed, curved 3-manifold, and cosmologists do, but there is no more proof that this model is the correct one than there was for the model with the earth at the center of the universe. Whether we determine that light propagates through 4-space in straight lines, or only on geodesic curves along 3-manifolds, we can only determine ''by looking'' that the space ''near'' us is resolutely 3-dimensional (not admitting the construction of four mutually perpendicular axes, only three). When we look out very far, at distant galaxy clusters for example, we have no way of determining whether we are looking through three dimensional space or four dimensional space. All those distant objects we see ''might'' lie in the same 3-manifold (perhaps on the same rough 3-sphere) that we do, but why should they have to? Might they not lie on separate 3-sphere soap bubbles, vastly distant from ours, whether or not all the soap bubbles had a common origin at one place and time? When we consider the ways in which particles propagating at the speed of light might reach us, considering that we ourselves are formations of particles propagating at the speed of light (all together in almost the same direction), it is clear that we ought not to expect to be overtaken by such particles emanating from stars on the opposite side of our own 3-manifold (from our antipodes, so to speak), because even such a particle redirected exactly backwards along the proper time axis of a star at our antipodes could only follow us along our opposite-direction proper-time axis at a fixed distance forever, never overtaking us, as we travelled in the same direction through 4-space at the same speed <math>c</math> forever. Or, since our own path through 4-space is a helical one (as we are engaged in numerous concentric orbits), if the pursuing particle's path through 4-space were a straighter one, it might in principle overtake us eventually, but probably not in our actual experience, and never as a particle moving relative to us at nearly the speed <math>c</math>. Therefore we should not expect to receive such particle radiation from the backside stars of our own 3-manifold. === The speed of light === So far, however, these considerations can apply only to mass particle radiation, not to light signals, since we have not yet described how light particles (photons) propagate through 4-space. We have suggested that elementary rigid objects propagate themselves by discrete Coxeter transformations,{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} </blockquote>|name=transformations}} that atomic mass particles are elementary rigid objects in some sense, and that all such particles are transforming at a constant rate <math>c</math> in various directions through Euclidean 4-space. But so far, the motion of mass particles is the only kind of motion we have described; we have not given an account of the nature of light signals, or the manner of their propagation, except to observe that light signals propagate through 4-space ''faster'' than <math>c</math>, as a consequence of the fact that they are observed to propagate through ''3-space'' at speed <math>c</math>. How much faster than <math>c</math> must photons be traveling through 4-space, since they appear to be traveling at <math>c</math> relative to two observers: one in the reference frame of the electron emitting the photon, and one in the reference frame of the electron absorbing the photon? If both observers are themselves traveling through 4-space at speed <math>c</math> as we have stipulated, then even in the case where their direction through 4-space is the same (they are at rest with respect to each other in the same reference frame), a photon that passes between them must travel at speed <math>\sqrt{2}c</math> if it makes the trip in a straight line, or even faster if it zig-zags in some fashion. Fortunately, this requirement is not at all paradoxical, since in a system of particles translating themselves through 4-space at the rate of <math>c</math> transforms per unit time, some things ''do'' move faster than speed <math>c</math>. A transforming rigid object with a translational motion at rate <math>c</math> may simultaneously have an orthogonal rotational motion at rate <math>c</math>, such that its component parts (e.g. each vertex of a rotating-translating 4-polytope) may displace themselves in 4-space ''more'' than one object-diameter in each discrete transformation; the combined rotating-and-translating velocity through 4-space of a ''component'' may be as much as twice the translational velocity of the whole rigid object, <math>\sqrt{4}c</math> rather than <math>\sqrt{1}c</math>, the [[W:8-cell#Radial equilateral symmetry|diagonal of an atomic unit 4-cube]] rather than its edge length. But the component points of such a rotating rigid object are all traveling in different directions at any instant, and the combined motion of the object as a whole cannot be other than <math>c</math>. Therefore a propagating light signal (a photon) is not a rigid atomic object, but some propagation of one of its component parts. Of course this agrees perfectly well with our understanding of photons as emissions of electrons, even if electrons are themselves rigid atomic objects translating themselves through 4-space at the rate of <math>c</math> transforms per unit time. The only "paradox" is linguistic in nature: <math>c</math> is not the "speed of light", it is the speed of matter (all mass-carrying particles) through 4-space. The actual ''speed of photons'' through 4-space is <math>2c</math>, as opposed to their observed speed through 3-space of <math>c</math>. === ... === A light signal (photon) propagates at speed c relative to either the emitting or the absorbing inertial reference frame (which reference frames are themselves in motion at speed c relative to their common 4-space stationary reference frame, but perhaps in different directions through that 4-space). ==== How soap bubble 3-manifolds behave in 4-space ==== 120 similar 2-sphere soap bubbles (spherical dodecahedra) tiling the 3-sphere, meeting at 120 degrees three-around each edge, and four at each vertex.{{Sfn|Stillwell|2001|p=24|loc=Figure 7. Soap bubble 120-cell}} === Atmospheric 3-membrane === What if the 3-space we observe (the visible universe) were filled with a gas, contained in some manner within the (thin) 3-membrane? Like the (thin) atmosphere of the earth (in a 3-dimensional analogy). In fact it is gas-filled, if the 4-space inside and outside the 3-membrane is empty. That difference is precisely what defines the 3-membrane: it is the 4-space which is not empty. It is not continuously full, as it is a cloud of discrete particles like a gas, and the density of particles is very low in most places, but within the 3-membrane it is never zero. The 3-membrane(s) is the surface(s) of 4-polytope(s) with a very large number of vertices (a function of the number of atoms in the universe, if those vertices are enumerated in some manner that includes the universe's plasma matter), and as in any 4-polytope all its elements lie on a 3-dimensional surface (albeit in the case of plasma on a rather inconsequential and rapidly changing 3D surface). === Configurations in the 24-cell === [[File:Reye configuration.svg|thumb|Reye's configuration 12₄16₃ of 12 lines (3 orthogonal groups of 4) intersecting at 16 points.]] The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye’s configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} In its most elemental expression, Reye's configuration is a set of 12 lines which intersect at 16 points, forming two disjoint cubes.{{Efn|The basic expression of Reye's configuration 12₄16₃ of 12 lines and 16 points occurs 12 times in the 24-cell, as the 16 vertices of two opposite (completely orthogonal) cubical cells (in one of the 3 inscribed 8-cells), and the 12 geodesic straight lines (hexagonal great circles) on which the cubes' parallel edges lie (orthogonally in the 3 dimensions of 3-sphere space, embedded in a Euclidean space of 4 orthogonal dimensions).}} It has multiple expressions in the 24-cell.{{Efn|An expression of Reye's configuration 12₄16₃ with respect to isoclinic rotations of the rigid 24-cell: 12 points are reached by 4 half hexagram isoclines in each left ''or'' right 360 degree isoclinic rotation characteristic of the 24-cell, and each of the 16 half hexagram isoclines in the left ''and'' right 360 degree rotations reaches 3 points.}} If the proton and neutron are indeed the hybrid fibrations we have identified, on that assumption we can count the number of valid protons and neutrons that can coexist in the same 24-cell at the same time without colliding, and the total number of distinct configurations of multiple nucleons (the number of distinct nuclides) that a single 24-cell can contain. This will not limit the number of distinct nuclides which can exist, because the 24-cell can be compounded in four-dimensional space in several different ways, but it will quantify the number of nuclides which can occupy the first nuclear shell. We might ask of any less minimal configuration of rotations in a single 24-cell whether it in fact corresponds to an integral number of nucleons occupying the same nuclear shell. A valid ''maximal'' hybrid configuration of rotations in the 24-cell would have the largest number of moving vertices possible without collisions (perhaps 24).{{Efn|The valid maximal configurations do have 24 moving vertices. They include a configuration with 12 vertices rotating ''within'' each of the three 16-cells on 4 octagram isoclines of the same chirality (while remaining within 4 Clifford parallel moving square planes), and visiting all 8 vertices of each 16-cell in each double revolution. The same maximal configuration also has 12 vertices rotating ''among'' the three 16-cells on 4 hexagram isoclines (two left-right pairs) while remaining within 4 Clifford parallel moving hexagonal planes, and visiting all 24 vertices of the 24-cell in each double revolution.}} ====...==== If the particle energies are to be described as the angular momentum of isoclinic rotations of some kind, it is noteworthy that square isoclinic rotations will describe 16-cells, but not (by themselves) the 24-cell. Square isoclinic rotations are associated with the ''internal'' geometry of 16-cells: they are the [[W:16-cell#Rotations|characteristic rotations of the 16-cell]]. The chords of the single [[W:16-cell#Helical construction|isocline of a square isoclinic rotation]] (left or right) are the four orthogonal axes of the 16-cell (enumerated twice), and the isocline is a helical circle passing through all 8 vertices of the 16-cell. A left-right pair of square rotations covers all the elements of the 16-cell (including its 18 great squares and its 16 tetrahedral cells). Thus the square isoclinic rotations say all there is to say about the internal geometry of an individual 16-cell, but they have nothing at all to say about how three 16-cells combine to form a 24-cell. The hexagonal isoclinic rotations, the [[W:24-cell#Isoclinic rotations|characteristic rotations of the 24-cell]], do describe the whole 24-cell. Their hexagram isoclines wind through all three 16-cells, their {{radic|3}} chords connecting the corresponding vertices of pairs of disjoint 16-cells. If vertices moving in hexagonal isoclinic rotations are what carries the energy binding the three quarks together (at least in the case of the neutron), each neutron would require at least two vertices moving in hexagonal isoclinic orbits, for the following reason. We attributed electric charge to the chirality of isospin generally, so we may expect hexagonal isoclinic rotations to contribute to the nucleon's total electric charge, even though they are not intrinsic parts of the three quarks, but rather parts of the whole 24-cell. Since the hexagonal orbits span the three 16-cells equally, the contributions of their moving vertices must be of neutral charge overall. Therefore the neutron must possess pairs of left and right hexagonal isoclinic rotations: minimally, one vertex moving on a left hexagram isocline, and one moving on a right hexagram isocline, which cancel each other because they have exactly opposite isospin. They must be a left and a right isocline from the same fiber bundle, corresponding to the left and right rotations of the same set of Clifford parallel invariant hexagonal planes of rotation. Such a pair is a valid kinematic rotation, because left and right hexagram isoclines of the same fibration do not intersect, and so can never collide.{{Efn|Most, but not all, left and right pairs of isoclinic rotations have isocline pairs which are Clifford parallel and visit disjoint vertex sets. The exception is left and right square isoclinic rotations. Their left and right [[W:16-cell#Helical construction|octagram isoclines]] ''do'' intersect, and they each visit the same set of 8 vertices.}} ====...==== The chiral pair of hexagonal rotations combined with the various square rotations will be a valid hybrid rotation provided no vertex in a hexagram orbit ever collides with a vertex in an octagram orbit. The octagram orbits visit all 8 vertices of the 16-cell in which they are confined. In an up quark with the minimum two moving vertices there are 6 empty vertices at any moment in time, leaving room for cross-traffic. The moving vertex on a hexagram isocline (of which minimally there will be two, a left and a right) will intersect each 16-cell in two places (not antipodal vertices) at different times. If the two moving vertices are antipodal (a moving axis), they will intersect each 16-cell in two axes at different times. There are valid configurations of this set of ''minimal'' hybrid rotations. In up-down-up proton configurations, the 7 moving vertices may be chosen in various ways that avoid collisions. Similarly in down-up-down neutron configurations, there are 6 moving vertices and various valid configurations. These minimal configurations could be are protons and neutrons, or they could be fractional parts of whole nucleons, less minimal configurations of the same kind of hybrid rotation with more moving vertices. These minimal hybrid rotations fall short of the full fibration symmetry of an ordinary isoclinic rotation in various ways. The hexagonally-rotating vertices visit only 12 of the 24 vertices once.{{Efn|Each hexagonal fibration has a right isoclinic rotation on 4 Clifford parallel right hexagram isoclines, and a corresponding left rotation on 4 Clifford parallel left hexagram isoclines, in the same set of 4 Clifford parallel invariant hexagonal planes. The right and left rotations reach disjoint sets of 12 vertices.}} The two square-rotating vertices in each up 16-cell visit all 8 vertices twice; the one in each down 16-cell visits all 8 vertices once. The two hexagram vertices each rotate through three 16-cells, so even with the best synchronization, there will be an oscillation in the total number of moving vertices in each 16-cell at any one time. In these minimal configurations, the 16-cells and the 24-cell would be strangely unbalanced objects. More generally, any such balance would require solutions to the n-body problem for 7 and 6 bodies, respectively. We could attempt to remedy these deficiencies by adding more moving vertices on more isoclines, seeking to make nucleons which are ''hybrid fibrations'' with at least one moving vertex on each isocline of each kind of fibration. This more balanced configuration with complete fibrations can be achieved with 6 vertices moving on hexagram isoclines, synchronized so that there are two of these moving vertices in each 16-cell at once. The proton (or neutron) will be a valid hybrid fibration of the 24-cell if it is synchronized to avoid collisions, with 6 vertices moving over all 24 vertices on isoclines of hexagonal fibrations (3 on left isoclines, and 3 on right isoclines), and 5 (or 4) vertices moving over all 24 vertices on isoclines of square fibrations (4 (or 2) on right isoclines, and 1 (or 2) on left isoclines). The up 16-cells already have two vertices moving on octagram isoclines in the minimal configuration, but the down 16-cells have only one. We can add two vertices moving on a chiral pair of isoclines without breaking the charge balance. Adding two such left-right pairs to each down 16-cell leaves the down 16-cell with two up+right and three down−left vertices. It could decay into an up 16-cell (two up+right vertices) by losing the three down−left vertices with a combined unit negative charge, like the electron emitted during [[W:beta decay|beta minus decay]]. This configuration of a single stable proton (or single unstable neutron) has 9 (or 12) vertices moving on isoclines of square fibrations: 6 (or 6) on right isoclines, and 3 (or 6) on left isoclines. The fact that a proton and neutron form a stable nuclide suggests that they can occupy the same 24-cell together, where their moving vertices combine to stabilize the down 16-cell. This is possible because there is enough room in each 16-cell for it to contain the essential moving vertices of both an up and a down quark at the same time: two vertices moving on an up+right isocline, plus one vertex moving on a down−left isocline provided it is not the complement of either of the right rotations (because the left-right pair of the same distinct rotation have exactly opposite isospin and would cancel each other's charge contribution). This [[W:deuterium|deuterium]] configuration of a stable proton-neutron pair has 9 vertices moving on isoclines of square fibrations: 6 on right isoclines, and 3 on left isoclines. It could be created by [[W:beta plus decay|beta plus decay]] when two protons are forced to occupy the same 24-cell, during the first step in the [[W:proton-proton chain|proton-proton chain]] of [[W:Stellar nucleosynthesis|nucleosynthesis]]. == Wiki researchers == [[W:User:Cloudswrest/Regular_polychoric_rings]], A.P. Goucher (https://cp4space.wordpress.com/2012/09/27/good-fibrations/) [[W:User:Tomruen/Uniform_honeycombs]] [[W:User:PAR]] physicist (stat. mech.) ([[wikipedia:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] in [[wikipedia:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]) [[W:User:DGG]], David Goodman (wikipedia admin, librarian, expert on scientific publishing and open access to published research, first amendment absolutist) [[User:Fedosin|S.Fedosin]] e.g. [[SPФ symmetry]], [[Scale dimension]], '''Model of Gravitational Interaction'''<ref>{{Cite web|url=http://sergf.ru/mgen.htm|title=Model of Gravitational Interaction in the Concept of Gravitons|website=sergf.ru|access-date=2019-05-11}}</ref>, [[Hydrogen system]], [[Metric theory of relativity]] - perhaps begin here<ref>http://vixra.org/pdf/1209.0110v1.pdf</ref> === Fibrations === Currently titled '''Visualization''', perhaps this section could be renamed and multiple references given for the [[W:Hopf fibration]]{{Efn|On a 2-sphere (globe), if you go off in any direction and keep going straight you eventually arrive back at your starting point. Same with the 3-sphere, except you are no longer restricted to a plane, you can go off in any 3-dimensional direction. For the 2-D case all great circles intersect. You can avoid this for the 3-D case. Step away from your initial starting point and go off in a new direction. You want to pick this direction so that you don't intersect the previous geodesic. To this end you have to give your new direction a little "skew" so that your new starting direction is not exactly parallel, and out of the plane, to your old direction. This avoidance of intersection causes the two loops/geodesics to spiral around each other and/or interlock. For the Hopf fibration the farther away you are from the initial starting point, the more "skew" you add. When you are 90 degrees away, you add 90 degrees of skew. This is the most extreme case and you have two interlocking rings passing through the midpoint of the other ring. With this construction you can parameterize the whole 3-sphere, with no two rings ever touching each other. ---- [[W:User:Cloudswrest]] in [[W:Talk:Hopf fibration#Untitled|Talk:Hopf fibration]]|name=|group=}} re: the 24-cell. But beware [[W:User:Cloudswrest/Regular polychoric rings|similar material]] by [[W:User:Cloudswrest]] and [[W:User:Tomruen]] was deleted from the [[W:Hopf fibration]] article once for lack of references. {{Efn|This section was deleted for lacking references. I had been adding some graphics. I copied the section to User:Tomruen/Regular polychoric rings. Tom Ruen (talk) 04:01, 15 November 2014 (UTC) I started the "Discrete Examples" section because of the 120-cell. I consider it a perfect example of the Hopf fibration in a different context. A "physical" example. People might not "get it" when given equations, or theory, or even a continuum picture, but seeing the 120-cell example might provide an "aha moment"! You can SEE it in the Todesco Youtube video of the 120-cell. The other face-to-face cases quickly became obvious. Then somebody mentioned, without any references, the BC helices in the 600-cell, and all the tet based polytope fibrations fell into place also. For the most part math articles have been a pretty safe subject to edit as it's objective, self documenting, except for very esoteric stuff, and people who don't know anything about it are usually uninterested, unlike articles on subjects like say, Martin Luther King, or Hitler, or date rape, where all the social justice warriors and polemicists come out to play. When Eppstein first complained about references over a year ago I did a web search. There are bits and pieces on various web sites and blogs, including some on John Baez's blog, but I could not find any coherent full coverage of the subject, which in any case is pretty obvious to interested parties. But I do know that making snide and sarcastic remarks on the rather competent and prolific work of a Wikipedia math illustrator is over the top. Cloudswrest (talk) 02:00, 17 November 2014 (UTC)}} [https://cp4space.wordpress.com/2012/09/27/good-fibrations/ Goucher] [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] [http://members.home.nl/fg.marcelis/mathemathics.htm Marcelis] The first illustration below shows a torus surface on which 4 equidistant circles lie, each having 4 equidistant points that are 4 of the 16 vertices of a hypercube in stereographic projection. On the vertical line and the circle lie 4 equidistant points each, completing the 16 vertices of the hypercube to the 24 vertices of a 24-cell. === Curiosities === https://physics.info/motion/ https://hexnet.org/ [[w:Double bubble conjecture|Double bubble conjecture]] http://vixra.org/pdf/1812.0482v1.pdf perhaps follows Steinbach's polygonal chord relationships === Review === [https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/ Baez] [http://eusebeia.dyndns.org/4d/bi24dim600cell Who is this?] [http://www.cs.utah.edu/~gk/peek/600slice/index.html Peek software] Who is this? http://eusebeia.dyndns.org If identified perhaps could put under External links on some 4-polytope pages. Especially [http://eusebeia.dyndns.org/4d/vis/vis 4D Visualization] [http://members.home.nl/fg.marcelis/ Marcelis] also [https://fgmarcelis.wordpress.com/ Macelis's other website] === Communities === [http://hi.gher.space/forum/ Higher Space Forum] == 4-space generally == === Dimensional analogy opinion === {{Efn|A [[W:Four-dimensional space#Dimensional analogy|dimensional analogy]] is not a metaphor that we are free to adopt or replace, like the conventional names of the 4-polytopes. The 600-cell is the unique 4-dimensional analogue of the icosahedron in a precise mathematical sense. The symmetry group of the 600-cell is only sometimes called the [[W:Binary icosahedral group|binary ''icosahedral'' group]] (by metaphorical analogy), but the dimensional relationship between the 600-cell and the icosahedron which the operations of the group capture is a mathematical fact (a dimensional analogy). It is not a mistake to call the 600-cell the hexacosichoron or the 4-120-polytope or any other reasonably analogous name we may invent, but it would be a mathematical error to misidentify the 600-cell as the analogue of some other polyhedron than the icosahedron.{{Efn|It is important to distinguish ''dimensional'' analogy from ordinary ''metaphorical'' analogy. ''Dimensional analogy''{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process that can function as a guide to proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction.|name=dimensional analogy}}|name=4-dimensional analogue of the icosahedron|group=}} {{Efn|It is a mistake to confuse the finite mathematics of ''dimensional analogy'' with the infinite art of ''metaphorical analogy''. Dimensional analogy{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process, like a proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction. [https://www.npr.org/books/titles/138359394/what-we-believe-but-cannot-prove-todays-leading-thinkers-on-science-in-the-age-o I believe, but I cannot prove], that there is but one ''correct'' dimensional analogy in every instance; moreover, there is ''always'' that one correct dimensional analogy in every instance (though it may well not have been discovered yet).|name=dimensional analogy}} === Words === [http://os2fan2.com/gloss/index.html The polygloss] Wendy Krieger's glossary of higher-dimensional terms. === Math === The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] {1, ''ξ''₁, ''η'', ''ξ''₂} [Coxeter 1973 p. 216]. Formulas for the conversion of Cartesian coordinates to Hopf coordinates: https://marc-b-reynolds.github.io/quaternions/2017/05/12/HopfCoordConvert.html In 3D every displacement can be reduced to a single-rotation combined with a translation (a screw-displacement). In 4D every displacement can be reduced to either a double rotation or a a single rotation combined with a translation (a 3D screw-displacement).[Coxeter 1973 p 218][[File:Pythagorean theorem - Ani.gif|thumb|caption=(3,4,5) is the smallest [[W:Pythagorean triple|Pythagorean triple]] (a [[W:Special right triangle#Side based|special right triangle]])]] The [[W:Icosian|Icosians]] and [[W:Quaternions]] generally, the "3<small>{{Sfrac|1|2}}</small>-dimensional coordinates of projective 4-space [[https://books.google.com/books?id=5-UlBQAAQBAJ&pg=PA207&lpg=PA207&dq=icosian+ring+golden+field&source=bl&ots=_bR1ndRqQ0&sig=ACfU3U1ePHtwHLEfyG5IC0bQr8Ur736hPw&hl=en&sa=X&ved=2ahUKEwjUiKSPzoriAhWEjp4KHYclD6EQ6AEwCXoECAgQAQ#v=onepage&q=icosian%20ring%20golden%20field&f=false|from google books]] Application of quaternions and projective space generally in the vertex figure space (the curved boundary space of a 4-polytope from the inside). [http://eusebeia.dyndns.org/4d/genrot.pdf Formula for Vector Rotation in Arbitrary Planes] [https://www.facebook.com/Formule.byBNF/posts/httpeusebeiadyndnsorg4dindex/10151772568869171/ Viviani's Theorem] In an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the altitude of the triangle [https://fgmarcelis.wordpress.com/ Macelis's other website] on the geometrical mathematics of physics. === Truncation === What does truncation look like from vertex space (the curved 3-manifold)? Vertices are removed like voids carved out of the interior of the 3-space, or rather, the location of the 3-manifold moves in the 4th direction. How can we imagine observing this (as a continuous process) from the inside of the 3-space? == 120-cell == "Who ordered that?"{{Efn|As the Nobel laureate physicist [[W:Isidor Isaac Rabi|I. I. Rabi]] famously quipped about the unanticipated muon, "Who ordered that?".}} [http://eusebeia.dyndns.org/4d/news2014q1 Omnitruncated 120-cell] - the largest uniform convex 4-polytope. [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] and [https://homepages.warwick.ac.uk/~masgar/Maths/quintessence.pdf Schleimer] A first way to understand the combinatorics of the 120–cell is to look at the layers of dodecahedra at fixed distances from the central dodecahedron. A second way to understand the 120–cell is via a combinatorial version of the [[W:Hopf fibration]]. === Falsified theory === To generate the 120-cell from the 600-cell, it is sufficient to rotate the 600 tetrahedra once, through five positions (either the left-handed or the right-handed chiral rotation). Both chiral rotations are not required, because each 5-click rotation by itself generates all the cells of five ''disjoint'' 600-cells, which together comprise all the vertices of the 120-cell and all ten 600-cells. In other words, the two ways to pick five disjoint 600-cells (out of the ten ''distinct'' 600-cells) correspond to the two sets of opposing tetrahedra in each dodecahedron. FALSIFIED{{Sfn|van Ittersum|2020|loc=§4.3.4 Quaternions with real part 1/2 in each 24-cell in the 600-cell 2I|pp=85-86}} === Dodecahedron coordinates === The red vertices lie at (±φ, ±{{sfrac|1|φ}}, 0) and form a rectangle on the ''xy''-plane. The green vertices lie at (0, ±φ, ±{{sfrac|1|φ}}) and form a rectangle on the ''yz''-plane. The blue vertices lie at (±{{sfrac|1|φ}}, 0, ±φ) and form a rectangle on the ''xz''-plane. (The red, green and blue coordinate triples are circular permutations of each other.) === 30-tetrahedron rings are duals of Petrie polygons === [[W:Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons|Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons]] == 600-cell == === Rotations === In the 600-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at most 10 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[digon]], and the completely orthogonal fixed plane intersects 0 vertices, 2 vertices (a digon), 4 vertices (a square), or 6 vertices (a hexagon), respectively. === Misc === [[User:Dc.samizdat/600-cell]] The 120 vertices can be seen as the vertices of four sets of 6 orthogonal equatorial pentagons which intersect only at their common center.{{Efn|The edges of the 600-cell form geodesic (great circle) decagons. One can pick out six orthogonal decagons, lying (for example) in the six orthogonal planes of the 4-axis coordinate system. Being completely orthogonal, these decagons share no vertices (they 'miss' each other and intersect at only one point, their common center). Thus they comprise 60 distinct vertices: half the vertices of the 600-cell. By symmetry, the other 60 vertices must occur in an exactly similar (congruent) configuration as another set of six orthogonal decagons (rotated isoclinically with respect to the first set).|name=orthogonal decagons}}{{Efn|Each decagon in the orthogonal set of 6 must share two vertices (a common diameter) with each decagon to which it is not orthogonal (namely, the 66 decagons not in the set). So each set of 6 (orthogonal) decagons populates vertices in 66 (other) decagons. There are 12 sets of 6 orthogonal decagons.}} {{Efn|For any fixed value of 𝜂, we have a 𝜉_''i'' decagon and a 𝜉_''j'' decagon with disjoint vertex sets, because they are completely orthogonal. Conversely, the 6 decagons which intersect at each vertex cannot be mutually orthogonal, and each must have a different value of 𝜂.}} The 600 tetrahedral cells can be seen as the result of a 5-fold subdivision of 24 octahedral cells yielding 120 tetrahedra, in a compound made of 5 such subdivided 24-cells (rotated with respect to each other in angular units of {{sfrac|𝜋|5}}). The 600-cell's edge length is ~0.618 times its radius (the 24-cell's edge length). This is 𝚽, the smaller of the two golden sections of √5. Its reciprocal, the larger golden section, is φ = 1.618. A {{radic|5}} chord will not fit in a polytope of unit radius ({{radic|4}} diameter), but both of its golden sections will fit, and both occur as vertex chords of the unit-radius 600-cell: the smaller 𝚽 as its edge length, and the larger φ as the chord joining vertices that are 3 edge lengths apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 4 sets of) [[wikipedia:24-cell#Hexagons|4 orthogonal hexagons]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 such disjoint sets of 4 orthogonal hexagons will account for all 120 vertices. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 3 sets of) [[wikipedia:24-cell#Squares|6 orthogonal squares]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 disjoint sets of 6 such orthogonal squares will account for all 120 vertices. Notice the pentagon inscribed in the decagon. Its {{radic|1.𝚫}} edge chord falls between the {{radic|1}} hexagon and the {{radic|2}} square. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges, evidently dodecahedra), which falls between the 24-cells' envelopes of octahedra (made of {{radic|1}} hexagon edges) and the 8-cells' envelopes of cubes (made of {{radic|2}} square edges). Consider also the {{radic|2.𝚽}} = φ and {{radic|3.𝚽}} chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell.{{Efn|1=The {{radic|2.𝚽}} = <big>φ</big> and {{radic|3.𝚽}} chords produce irregular interior faces and cells, since they make isosceles great circle triangles out of two chords of their own size and one of another size.|name=isosceles chords|group=}} The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell, it also inserts new features deep in the interstices of the [[wikipedia:24-cell#Constructions|24-cell's interior]] structure (which it inherits in full, compounds five-fold, and then elaborates on). evidently the 600-cell has dodecahedra in it golden triangles{{Efn|A [[W:Golden triangle|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ration|golden ratio]] to the distinct side ''b'': : {{sfrac|a|b}} <nowiki>=</nowiki> ϕ <nowiki>=</nowiki> {{sfrac|1 + {{radic|5}}|2}} <nowiki>≈</nowiki> 1.618 It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center.<br> The vertex angle is: : <nowiki>𝛉 = arccos(</nowiki>{{sfrac|ϕ|2}}<nowiki>) = </nowiki>{{sfrac|𝜋|5}}<nowiki> = 36°</nowiki> so the base angles are each {{Sfrac|2𝜋|5}} <nowiki>=</nowiki> 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=golden triangle}} ==== Rotations ==== {{Efn|This is another aspect of the same pentagonal symmetry which permits the partitioning of the 600-cell into [[#Icosahedra|icosahedral clusters]] of 20 cells and clusters of 5 cells.}} Each isoclinic rotation occurs in two chiral forms: there is a Clifford parallel 24-cell to the ''left'' of each 24-cell, and another Clifford parallel 24-cell to its ''right''. The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. ==== Central planes ==== All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart in the 600-cell), hexagon planes ({{sfrac|𝜋|3}} apart in each of 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart in each of 75 inscribed 16-cells). In a 4-polytope, two different central planes may intersect at a common diameter, as they would in 3-space, or they may intersect at a single point only, at the center of the 4-polytope. In the latter case, their great circles are [[W:Clifford parallel|Clifford parallel]].{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space.}} Completely orthogonal{{Efn|name=completely orthogonal planes}} great circles are an example of Clifford parallels, but we can also find non-orthogonal central planes which intersect at only a single point. {{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Because they share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3-dimensions; in fact they are connected like adjacent links in a chain, each passing through the other (without intersecting at any points), forming a [[W:Hopf link|Hopf link]]. === Cayley’s Factorization of 4D Rotations === Any rotation in ℝ⁴ can be seen as the composition of two rotations in a pair of orthogonal two-dimensional subspaces. When the values of the rotation angles in these two subspaces are equal, the rotation is said to be isoclinic. Cayley realized that any rotation in ℝ⁴ can be factored into the commutative composition of two isoclinic rotations.<ref>{{Cite journal|last=Perez-Gracia|first=Alba|last2=Thomas|first2=Federico|date=2016-05-14|title=On Cayley’s Factorization of 4D Rotations and Applications|url=http://dx.doi.org/10.1007/s00006-016-0683-9|journal=Advances in Applied Clifford Algebras|volume=27|issue=1|pages=523–538|doi=10.1007/s00006-016-0683-9|issn=0188-7009}}</ref> The elements of the Lie group of rotations in four-dimensional space, SO(4), can be either simple or double rotations. Simple rotations have a fixed plane (a plane in which all the points are fixed under the rotation), while double rotations have a single fixed point only, the center of rotation. In addition, double rotations present at least a pair of invariant planes that are orthogonal. The double rotation has two angles of rotation, α1 and α2, one for each invariant plane, through which points in the planes rotate. All points not in these planes rotate through angles between α1 and α2. Isoclinic rotations are a particular case of double rotations in which there are infinitely many invariant orthogonal planes, with same rotation angles, that is, α1 = ±α2. These rotations can be left-isoclinic, when the rotation in both planes is the same (α1 = α2), or right-isoclinic, when the rotations in both planes have opposite signs (α1 = −α2). Isoclinic rotation matrices have several important properties: # The composition of two right- (left-) isoclinic rotations is a right- (left-) isoclinic rotation. # The composition of a right- and a left-isoclinic rotation is commutative. # Any 4D rotation can be decomposed into the composition of a right and a left-isoclinic rotation. Hence both form maximal and normal subgroups. Their direct product is a double cover of the group SO(4), as four-dimensional rotations can be seen as the composition of rotations of these two subgroups, and there are two expressions for each element of the group. === 30-gon geodesic === The [[w:600-cell#Geodesics|30-gon vertex-less geodesic of the 600-cell]] reminds me of another remarkable observation about the central axis of the B-C helix made many years ago by the dutch software engineer and geometry experimenter Gerald de Jong,{{Efn|I don't recall de Jong ever writing about 4-dimensional polytopes, but he has a large body of work experimenting with physical and virtual models of geodesics and especially tensegrity structures.}} on a long-extinct email list called ''Synergetics'' that mostly featured discussions of Buckminster Fuller{{Efn|Buckminster Fuller never quite got his mind around 4-polytopes, despite knowing Coxeter, but much of what he observed about the 3-polytopes is directly relevant to the 4-polytopes and original; he had splendid intuition. For example in his obsession with the cuboctahedron (which he called the "vector equilibrium") he was probably the first to sense the real importance of the [[w:24-cell#Radially equilateral honeycomb|radially equilateral]] polytopes. Looking at the 24-cell and tesseract makes me sad for him that he never realized the fourth dimension has a ''regular'' vector equilibrium, one of ''two'' radially equilateral regular polytopes (the other of which is the hypercube!). Just as Fuller's studies and those he inspired (such as [[http://verbchu.blogspot.com/2010/07/ccp-and-hcp-family-of-structures.html%7Cthis]]) are often relevant to the 4-polytopes, the 4-polytopes now inspire new 3-dimensional inventions, such as new forms of Fuller's geodesic domes{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} memes. }} That list didn't extend to 4-polytopes; the geometry discussed there was about 3-dimensional objects, as it also tended to be on [[Magnus Wenninger]]'s ''Polyhedron'' email list. I can't find an archived copy of the email list with Gerald's post but as I recall he studied the B-C helix (Fuller called it the ''tetrahelix'') in 3 dimensions and observed that it had no single central axis, but rather three central axes that passed through each tetrahedron similarly, hitting the volume center of the tetrahedron and hitting two faces near but not at their center, like three holes punched in the face in a small equilateral triangle surrounding the face center. He called the tetrahedra pierced by the three central axes "tetrahedral salt cellars", a wonderfully evocative image and why I have remembered it (correctly, I hope). It is interesting to see that when the helix is bent in the fourth dimension into a ring, in addition to its period being rationalized and its helical edge-paths being straightened into geodesics, its three center axes also merge into one, which passes through a single point at the center of each face. === Golden triangles === The [[W:Golden triangle (mathematics)|golden triangle]] is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions. [[File:Golden_Triangle.svg|right|thumb|A golden triangle. The ratio a:b is equivalent to the golden ratio φ.]] [[w:Golden triangle|Golden triangles]] are found in the nets of several stellations of dodecahedrons and icosahedrons. Since the angles of a triangle sum to 180°, base angles are therefore 72° each.^[1] The golden triangle can also be found in a regular decagon, or an equiangular and equilateral ten-sided polygon, by connecting any two adjacent vertices to the center. This will form a golden triangle. This is because: 180(10-2)/10=144 degrees is the interior angle and bisecting it through the vertex to the center, 144/2=72.^[1] [[File:Kepler_triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the [[w:Golden_ratio|golden ratio]].]] A [[w:Kepler_triangle|Kepler triangle]] is a right triangle with edge lengths in a geometric progression in which the common ratio is √φ, where φ is the golden ratio,^[a] and can be written: , or approximately '''1 : 1.272 : 1.618'''.^[1] The squares of the edges of this triangle are in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between its short side and hypotenuse equal to the golden ratio.^[2] Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed:<blockquote>Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.^[3]</blockquote>Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza,^[4][5] making it a golden pyramid. === Golden chords === The ''golden chords'' demonstrate that <big>ϕ</big> is a circle ratio like <big>𝜋</big>, in fact:<br> : {{sfrac|𝜋|5}} = arccos ({{sfrac|ϕ|2}}) which is one decagon edge. Inversely:<br> : <big>ϕ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})<br> which can be seen from the arc length of the {{radic|2.𝚽}} = <big>ϕ</big> golden chord which is {{sfrac|3𝜋|5}}, but it was apparently discovered first without recourse to geometry.<ref>{{Cite web|title=Pi, Phi and Fibonacci|date=May 15, 2012|author=Gary Meisner |url=https://www.goldennumber.net/pi-phi-fibonacci/|postscript=: Robert Everest discovered that you can express <big>ϕ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series: <big>ϕ</big> = 1 – 2 cos (3𝜋/5)}}</ref> Phi is a circle ratio, like Pi Pi = 5 arccos (.5 Phi) Note: The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians. Note: Above formulas expressed in radians, not degrees Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows: 5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series: Phi = 1 – 2 cos ( 3 Pi / 5) Golden ratio of chords: Peter Steinbach * Golden Fields - DPF formula https://www.jstor.org or https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1997.11996494 * Sections Beyond Golden https://archive.bridgesmathart.org/2000/bridges2000-35.pdf Sacred cut (of octagon) https://archive.bridgesmathart.org/2011/bridges2011-559.pdf [[W:George Phillips Odom Jr.]] discovery of golden section in the mid-edge-bisectors of the tetrahedron (applies to the construction of the 600-cell from the 24-cell via truncation of its central tetrahedra) - also look at the other circle/triangle and sphere/tetrahedra relationships he discovered - footnote his relationship to Coxeter and Conway [[w:Intersecting chords theorem|Intersecting chords theorem]] === Nonconvex regular decagon === [[File:Golden_tiling_with_rotational_symmetry.svg|left|thumb|This '''[[Tessellation|tiling]]''' by '''[[Golden triangle (mathematics)|golden]]''' triangles, a regular '''[[pentagon]]''', contains a '''[[wikipedia:Stellation|stellation]]''' of '''[[Regular polygon|regular]] decagon''', the '''[[Schläfli symbol|Schäfli&nbsp;symbol]]''' of which is {10/3}.]] The length [[ratio]] of two inequal edges of a golden triangle is the [[golden ratio]], denoted <math>\text{by }\Phi \text{,}</math> or its [[Multiplicative inverse|multiplicative&nbsp;inverse]]: :<math> \Phi - 1 = \frac{1}{\Phi} = 2\,\cos 72\,^\circ = \frac{1}{\,2\,\cos 36\,^\circ} = \frac{\,\sqrt{5} - 1\,}{2} \text{.}</math> So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this [[Star polygon|star&nbsp;polygon]].{{Clear}} == 24-cell == {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} [[File:24-cell graph D4.svg|thumb]] Whoever this is has some beautiful illustrations of closely related polytopes e.g.<ref>http://eusebeia.dyndns.org/4d/rect24cell</ref> [http://members.home.nl/fg.marcelis/24-cell.htm Marcelis] including [http://members.home.nl/fg.marcelis/24celiso.htm#gem the gem of the modular universe] === Cuboctahedron === In the 24-cell, there are 16 great hexagons, and each has one pair of vertices (one diameter) which lies on a Cartesian coordinate system axis (in the vertex-up frame of reference). The 24-cell can be seen as the 24 vertices of four orthogonal great hexagons, each aligned with one of the four orthogonal coordinate system axes, and each contributing 6 disjoint vertices. In four dimensions, the four ''orthogonal'' planes do not intersect except at their common center. Each great hexagon does not share any vertices with the 3 other hexagons to which it is orthogonal,{{Efn|In four dimensions up to 6 planes through a common point may be mutually orthogonal. The 18 great squares of the 24-cell comprise three sets of 6 orthogonal planes. The 16 great hexagons, however, comprise four sets of just 4 orthogonal planes. In four dimensions there is both a symmetrical arrangement of 6 orthogonal planes, and a symmetrical arrangement of 4 orthogonal planes. We can pick out 6 orthogonal squares in the 16-cell, 8-cell, or 24-cell, but the symmetry of 4 orthogonal hexagons emerges only in the 24-cell.}} but it shares two vertices with each of the 12 other great hexagons to which it is ''not'' orthogonal. Four hexagonal geodesics pass though each vertex. In the cuboctahedron, there are four great hexagons, but they are not orthogonal. Each intersects with each of the others in two vertices (one diameter), and two hexagons pass through each vertex. (In three dimensions, two planes through a common point intersect in a line, whether they are orthogonal or not.) At most one of the four hexagons can have a pair of vertices (a diameter) which lies on a coordinate system axis. In such a frame of reference, the other two axes pass through the centers of a pair of opposing triangular faces, and through the centers of a pair of opposing edges, respectively. === Geometry === ==== Triangles ==== ...to be added:<br> If the dual of the [[W:24-cell#Squares|24-cell of edge length {{radic|2}}]] is taken by reciprocating it about its ''circumscribed'' sphere, another 24-cell is found which has edge length and circumradius {{radic|3}}, and its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows: :<math>(0, \pm 1, \pm 1, \pm 1) \in \mathbb{R}^4</math> The 4 orthogonal planes in which the 8 triangles lie are ''not'' orthogonal planes of this coordinate system. The triangles' {{radic|3}} edges are the ''diagonals'' of cubical ''cells'' of this coordinate lattice.{{Efn|For example: {{green|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> are the two opposing central triangles on the ''y'' axis (in this coordinate system of edge length {{radic|3}}).|name=|group=}} The 24 vertices are also the vertices of 96 ''other'' triangles of edge length {{radic|3}} that occur in 48 parallel pairs, in planes one edge length apart. Each plane sections the 24-cell through three vertices but does not pass through the center.{{Efn|...add coordinates example to existing note...|name=|group=}} ..see [[w:5-cell#Construction|5-cell#Construction]]: Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}}: :<math>\left( 1,1,1, \frac{-1}\sqrt{5} \right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,-1,1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0,0,0,\sqrt{5}-\frac 1\sqrt{5} \right)</math> ... obs: <br> Add coordinate examples to this footnote{{Efn|Each of these 96 triangular planes sections the 24-cell <small>{{sfrac|1|2}}</small> edge-length below a vertex, and <small>{{sfrac|1|2}}</small> edge-length above the center, measured from the center of the triangle, which is on a 24-cell diameter joining two opposite vertices. However, these paired parallel triangular planes are not orthogonal to the diameter line; they are inclined with respect to it. Each plane contains only one triangle (unlike the central hexagonal planes with their two opposing triangles), but they occur in co-centric sets of four, inclined different ways about the diameter line. The 96 triangles are inclined both with respect to the coordinate system's 6 orthogonal planes (the 6 perpendicular squares) and with respect to the hexagons. Each triangle contains one vertex from a square, and two from different hexagons. Thus their Cartesian coordinates take many different forms, but as examples: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> is one of the four face triangles with one vertex at the positive vertex on the ''y'' axis; and below is the opposite face of one of the two tetrahedra it is a face of, inclined about the negative part of the ''y'' axis: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> |name=non-central triangles}} === Rotations === The vertices of a convex 4-polytope lie on the [[W:3-sphere|3-sphere]], so alternatively they can be described using 4-dimensional spherical coordinates such as the [[w:N-sphere#Hopf_coordinates|Hopf coordinates]] {''r'', ''ξ''₁, ''η'', ''ξ''₂}, which reveal more structure. For the 24-cell of edge length and radius 1, the Hopf coordinates of its vertices can be obtained by permuting the three angle coordinates as follows: {1, ±{{sfrac|𝜋|3}}, ±{{sfrac|𝜋|2}}, ±{{sfrac|2𝜋|3}}}[[File:30-60-90.svg|thumb]] in 3D section of a 24-cell (of edge length 2) we can construct a tetrahedron with 2{{radic|3}} legs descending from a vertex (0,0,2) with its base plane triangle intersecting the 2-sphere at (x,y,-1), but only if we put the base vertices at distances apart less than 2{{radic|3}}, as a 2{{radic|2}} 3 3 isosceles triangle e.g. with these vertices: (-{{radic|2}},-1,-1) <--- 2{{radic|2}} --> ((+{{radic|2}},-1,-1)) <-- 3 --> (0, (+{{radic|3}}, -1) <-- 3 --> Apparently the base vertices of the tetrahedron are displaced out of this hyperplane in the 4th dimension so the base edges are foreshortened. ... and eight meet at the volume center of each tesseract cube{{Efn|The geometry of the tesseract cube volume centers is exactly the same as the vertices: a cubical vertex figure, in which four long diameters cross at the center. They are 24 interior vertices, arrayed as a 1/2 size 24-cell around the central interior vertex, at the midpoints of the 24 unit-length radii (which pass through opposite face centers of the vertex figure).}} {{Clear}} ==Kepler problem== [[File:Kepler-solar-system-2.png|thumb|Detailed view of the inner sphere of Kepler's Mysterium Cosmographicum.]]We are apt to be smug about the quaint mythological phantasies of our great forebears, as when we learn that Issac Newton worked for more than 30 years as an alchemist trying to turn base metals into gold, before he was appointed Chancellor of the Exchequer to preside over England's mint of sterling instead. [[W:Mysterium Cosmographicum|Kepler's astrolabe]] of Plato's holy solids looks that way to us, like a religious miracle the great astronomer hoped for, before he discovered the real symmetry in his three great conservation laws of motion. It should humble us, then, to find out that he was on to something deeper all along, and make us wonder all the more at his genius, that it was the SO(4) rotational symmetry he glimpsed, which generates those conservation laws by Noether's theorem, and also generates the 4-polytopes that Schläfli would discover on the 3-sphere in 4-space two and a half centuries after Kepler, who somehow imagined them from below, projected on the 2-sphere in 3-space. ==Laplace–Runge–Lenz vector== the following rescued from [[W:Laplace–Runge–Lenz vector]] version of 23:32, 27 November 2006 from which it was removed by WillowW (talk | contribs) at 23:46, 28 November 2006 (the Moebius transform was fun, but needs to go now) ===Intuitive picture of the rotations in four dimensions=== [[Image:Kepler_hodograph_family_transformed.png|thumb|right|280px|Figure 4: [[W:Circle inversion|Inversion]] in the dotted black circle of Figure 3 transforms the family of circular hodographs of a given energy ''E'' into a family of straight lines intersecting at the same point. Thus, the orbits of the same energy but different angular momentum can be transformed into one another by a simple rotation.]] The simplest way to visualize the particular symmetry of the Kepler problem is through its [[W:hodograph|hodograph]]s, the perfectly circular traces of the momentum vector (Figures 2 and 3). For a given total [[W:energy|energy]] ''E'', the hodographs are circles centered on the ''p_y''-axis, all of which intersect the ''p_x''-axis at the same two points, ''p_x=±p₀'' (Figure 3). To eliminate the normal rotational symmetry, the coordinate system has been fixed so that the orbit lies in the ''x''-''y'' plane, with the major semiaxis aligned with the ''x''-axis. [[W:Circle inversion|Inversion]] centered on one of the foci transforms the hodographs into straight lines emanating from the inversion center (Figure 4). These straight lines can be converted into one another by a simple two-dimensional rotation about the inversion center. Thus, all orbits of the same energy can be continuously transformed into one another by a rotation that is independent of the normal three-dimensional rotations of the system; this represents the "higher" symmetry of the Kepler problem. == Formatting idioms == :{{sfrac|1|2}}(±φ𝞍ϕ𝜙𝝓𝚽𝛷𝜱, ±1, ±{{sfrac|1|φ}}, 0). : :360^{\circ} :denoted <math>\tbinom{24}{4}</math>   == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book)}} * {{Cite book | last1=Coxeter | first1=H.S.M. | author1-link=W:Harold Scott MacDonald Coxeter | last2=du Val | first2=Patrick | author2-link=W:Patrick du Val | last3=Flather | first3=H.T. | last4=Petrie | author4-link=W:John Flinders Petrie | first4=J.F. | year=1938 | title=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]] | publisher=University of Toronto Studies (Mathematical Series) | volume=6 }} * {{Cite book|title=The Coxeter Legacy|chapter=Coxeter Theory: The Cognitive Aspects|last=Borovik|first=Alexandre|year=2006|publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|pp=17-43|ISBN=978-0821837221|url=https://www.academia.edu/26091464/Coxeter_Theory_The_Cognitive_Aspects}} * {{Cite journal | last=Miyazaki | first=Koji | year=1990 | title=Primary Hypergeodesic Polytopes | journal=International Journal of Space Structures | volume=5 | issue=3–4 | pages=309–323 | doi=10.1177/026635119000500312 | s2cid=113846838 }} * {{Cite book|url=https://link.springer.com/chapter/10.1007/978-981-10-7617-6_6|title=Nanoinformatics|last=Nishio|first=Kengo|last2=Miyazaki|first2=Takehide|publisher=Springer|year=2018|isbn=|editor-last=Tanaka|location=Singapore|pages=97-130|chapter=Polyhedron and Polychoron Codes for Describing Atomic Arrangements}} * {{Cite journal|last=Waegell|first=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|url=https://arxiv.org/abs/0911.2289v2|language=en|doi=10.1088/1751-8113/43/10/105304}} * {{Cite journal|last=Sadoc|first=Jean-Francois|date=2001|title=Helices and helix packings derived from the {3,3,5} polytope|journal=[[W:European Physical Journal E|European Physical Journal E]]|volume=5|pages=575–582|doi=10.1007/s101890170040|doi-access=free|s2cid=121229939|url=https://www.researchgate.net/publication/260046074}} {{Refend}} <references group="lower-alpha" /> lvdw6lzebcg944cllxyfnzzia5iyfhi 2691726 2691725 2024-12-13T01:15:31Z Dc.samizdat 2856930 /* 24-cell */ 2691726 wikitext text/x-wiki {{User sandbox}} == Polyscheme learning project == The Polyscheme learning project is intended to be series of articles on the regular polytopes and Euclidean spaces of 4 or more dimensions, that expand the corresponding series of Wikipedia encyclopedia articles to book length with textbook-like treatment of the material, additional learning resources and explanatory notes. == David Christie on Tolkien, to Ira Sandperl (emeritus) on Tolstoy == No doubt Tolkien's novels are not as profound as Tolstoy's with respect to our human condition, but Tolkien's address the more general condition of all peoples. He was the first novelist to write about the affairs of many peoples rather than just the comparatively parochial affairs of men. There is an expansive diversity to his interests, which shows his concern to transcend not just the patriarchy, but the humanitarchy. It disproves his critics' claims that his focus is not wide enough to include women, or people of color, or domains of experience (such as sex) which the critics feel are not sufficiently represented in his stories. == Regular convex 4-polytopes == {{W:Template:Regular convex 4-polytopes|wiki=W:}} {{Regular convex 4-polytopes|wiki=W:}} {| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes |- !align=right|[[W:Symmetry group|Symmetry group]] |align=center|[[W:Coxeter_group|A₄]] |align=center colspan=2|[[W:Hyperoctahedral_group|B₄]] |align=center|[[W:F4_(mathematics)|F₄]] |align=center colspan=2|[[W:H4_polytope|H₄]] |- !valign=top align=right|Name |valign=top align=center|[[W:5-cell|5-cell]]<BR> <BR> Hyper-<BR> [[W:Tetrahedron|tetrahedron]] |valign=top align=center|[[W:16-cell|16-cell]]<BR> <BR> Hyper-<BR> [[W:Octahedron|octahedron]] |valign=top align=center|[[W:8-cell|8-cell]]<BR> <BR> Hyper-<BR> [[W:Cube|cube]] |valign=top align=center|[[W:24-cell|24-cell]] |valign=top align=center|[[W:600-cell|600-cell]]<BR> <BR> Hyper-<BR> [[W:Icosahedron|icosahedron]] |valign=top align=center|[[W:120-cell|120-cell]]<BR> <BR> Hyper-<BR> [[W:Dodecahedron|dodecahedron]] |- !align=right|[[W:Schläfli symbol|Schläfli symbol]] |align=center|{3, 3, 3} |align=center|{3, 3, 4} |align=center|{4, 3, 3} |align=center|{3, 4, 3} |align=center|{3, 3, 5} |align=center|{5, 3, 3} |- !align=right|[[W:Coxeter-Dynkin diagram|Coxeter diagram]] |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} |- !valign=top align=right|Graph |align=center|[[Image:4-simplex t0.svg|120px]] |align=center|[[Image:4-cube t3.svg|121px]] |align=center|[[Image:4-cube t0.svg|120px]] |align=center|[[Image:24-cell t0 F4.svg|120px]] |align=center|[[Image:600-cell graph H4.svg|120px]] |align=center|[[Image:120-cell graph H4.svg|120px]] |- !align=right|Vertices |align=center|5 |align=center|8 |align=center|16 |align=center|24 |align=center|120 |align=center|600 |- !valign=top align=right|Edges |align=center|10 |align=center|24 |align=center|32 |align=center|96 |align=center|720 |align=center|1200 |- !valign=top align=right|Faces |align=center|10<BR>triangles |align=center|32<BR>triangles |align=center|24<BR>squares |align=center|96<BR>triangles |align=center|1200<BR>triangles |align=center|720<BR>pentagons |- !valign=top align=right|Cells |align=center|5<BR>tetrahedra |align=center|16<BR>tetrahedra |align=center|8<BR>cubes |align=center|24<BR>octahedra |align=center|600<BR>tetrahedra |align=center|120<BR>dodecahedra |- !valign=top align=right|[[W:Cartesian coordinates|Cartesian]]{{Efn|The coordinates (w, x, y, z) of the unit-radius origin-centered 4-polytope are given, in some cases as {permutations} or [even permutations] of the coordinate values.}}<BR>coordinates |align=center|{{font|size=85%|( 1, 0, 0, 0)<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})}} |align=center|{{font|size=85%|(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})<BR>([±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0])}} |align=center| |- !valign=top align=right|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf]]{{Efn|name=Hopf coordinates key|[[W:3-sphere#Hopf coordinates|Hopf spherical coordinates]]{{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.{{Sfn|Sadoc|2001|pp=575-576|loc=§2.2 The Hopf fibration of S3}} A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two completely orthogonal invariant planes{{Efn|The point itself (𝜉_''i'', 𝜂, 𝜉_''j'') does not necessarily lie in either of the invariant planes of rotation referenced to locate it (by convention, the ''wz'' and ''xy'' Cartesian planes), and never lies in both of them, since completely orthogonal planes do not intersect at any point except their common center. When 𝜂 {{=}} 0, the point lies in the 𝜉_''i'' "longitudinal" ''wz'' plane; when 𝜂 {{=}} {{sfrac|𝜋|2}} the point lies in the 𝜉_''j'' "equatorial" ''xy'' plane; and when 0 < 𝜂 < {{sfrac|𝜋|2}} the point does not lie in either invariant plane. Thus the 𝜉_''i'' and 𝜉_''j'' coordinates number vertices of two completely orthogonal great circle polygons which do ''not'' intersect (at the point or anywhere else).|name=reference planes of rotation}} which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} The 𝜉_''i'' and 𝜉_''j'' coordinates range over the vertices of completely orthogonal great circle polygons which do not intersect at any vertices. Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<BR> : w {{=}} cos 𝜉_''i'' sin 𝜂<BR> : x {{=}} cos 𝜉_''j'' cos 𝜂<BR> : y {{=}} sin 𝜉_''j'' cos 𝜂<BR> : z {{=}} sin 𝜉_''i'' sin 𝜂<BR> The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).<BR> The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], denoted SO(4), generates those polytopes. A rotation in 4D of a point {{math|{''ξ''_i, ''η'', ''ξ''_j<nowiki>}</nowiki>}} through angles {{math|''ξ''₁}} and {{math|''ξ''₂}} is simply expressed in Hopf coordinates as {{math|{''ξ''_i + ''ξ''₁, ''η'', ''ξ''_j + ''ξ''₂<nowiki>}</nowiki>}}.}} of the vertices are given as three independently permuted coordinates:<BR> :(𝜉_''i'', 𝜂, 𝜉_''j'')_𝑚<BR> where {<''k''} is the {permutation} of the ''k'' non-negative integers less than ''k'', and {≤''k''} is the permutation of the ''k''+1 non-negative integers less than or equal to ''k''. Each coordinate permutes one set of the 4-polytope's great circle polygons, so the permuted coordinate set expresses one set of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-space]] which generates the 4-polytope. With Cartesian coordinates the choice of radius is a parameter determining the reference frame, but Hopf coordinates are radius-independent: all Hopf coordinates convert to unit-radius Cartesian coordinates by the same mapping. {{Efn|name=Hopf coordinates conversion}} Unlike Cartesian coordinates, Hopf coordinates are not necessarily unique to each point; there may be Hopf coordinate synonyms for a vertex. The multiplicity 𝑚 of the coordinate permutation is the ratio of the number of Hopf coordinates to the number of vertices.}}<BR>coordinates |align=center|[[User:Dc.samizdat/sandbox#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |align=center|{{font|color=green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}</small>)₁}} |- !valign=top align=right|Long radius{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope.|name=polytopes ordered by size and complexity}} |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |- !valign=top align=right|Edge length |align=center|<small>{{sfrac|{{radic|5}}|{{radic|2}}}}</small> ≈ 1.581 |align=center|{{radic|2}} ≈ 1.414 |align=center|1 |align=center|1 |align=center|<small>{{sfrac|1|ϕ}}</small> ≈ 0.618 |align=center|<small>{{Sfrac|1|{{radic|2}}ϕ²}}</small> ≈ 0.270 |- !valign=top align=right|Short radius |align=center|{{sfrac|1|4}} |align=center|{{sfrac|1|2}} |align=center|{{sfrac|1|2}} |align=center|<small>{{sfrac|{{radic|2}}|2}}</small> ≈ 0.707 |align=center|<small>1 - ({{sfrac|{{radic|2}}|2{{radic|3}}φ}})²</small> ≈ 0.936 |align=center|<small>1 - ({{sfrac|1|2{{radic|3}}φ}})²</small> ≈ 0.968 |- !valign=top align=right|Area |align=center|<small>10•{{sfrac|{{radic|8}}|3}}</small> ≈ 9.428 |align=center|<small>32•{{sfrac|{{radic|3}}|4}}</small> ≈ 13.856 |align=center|24 |align=center|<small>96•{{sfrac|{{radic|3}}|8}}</small> ≈ 20.785 |align=center|<small>1200•{{sfrac|{{radic|3}}|8φ²}}</small> ≈ 99.238 |align=center|<small>720•{{sfrac|25+10{{radic|5}}|8φ⁴}}</small> ≈ 621.9 |- !valign=top align=right|Volume |align=center|<small>5•{{sfrac|5{{radic|5}}|24}}</small> ≈ 2.329 |align=center|<small>16•{{sfrac|1|3}}</small> ≈ 5.333 |align=center|8 |align=center|<small>24•{{sfrac|{{radic|2}}|3}}</small> ≈ 11.314 |align=center|<small>600•{{sfrac|1|3{{radic|8}}φ³}}</small> ≈ 16.693 |align=center|<small>120•{{sfrac|2 + φ|2{{radic|8}}φ³}}</small> ≈ 18.118 |- !valign=top align=right|4-Content |align=center|<small>{{sfrac|{{radic|5}}|24}}•({{sfrac|{{radic|5}}|2}})⁴</small> ≈ 0.146 |align=center|<small>{{sfrac|2|3}}</small> ≈ 0.667 |align=center|1 |align=center|2 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 3.907 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 4.385 |} == Scratch == 400 <math>\sqrt{5} \curlywedge (2-\phi)</math> hexagons In each hemi-icosahedron, 15 edges come from 10 disjoint 5-cells each contributing 4 edges to this hemi-icosahedron, and 3 hemi-icosahedra fitting together around each edge sharing it, as Grünbaum discovered they do. One other hemi-icosahedron fits against each of 10 hemi-icosahedron faces, and two other hemi-icosahedra fit around each of 15 opposite edge, all the same set of 11 hemi-icosahedra. == Coordinate systems on the 3-sphere == The four Euclidean coordinates for {{math|''S''³}} are redundant since they are subject to the condition that {{math|1=''x''₀² + ''x''₁² + ''x''₂² + ''x''₃² = 1}}. As a 3-dimensional manifold one should be able to parameterize {{math|''S''³}} by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as [[latitude]] and [[longitude]]). Due to the nontrivial topology of {{math|''S''³}} it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use ''at least'' two [[coordinate chart]]s. === Hopf coordinates of the regular convex 4-polytopes === As with Cartesian coordinates, there are multiple reference frames which give useful Hopf coordinates.{{Efn|name=Hopf coordinates}} One can choose any of the 4-polytope's great circle polygons for the 𝜉_''i'' coordinate to range over; then the 𝜉_''j'' coordinate will range over the vertices of whatever kind of great circle polygon lies orthogonal to the 𝜉_''i'' great circle plane. Note that the 𝜉_''j'' polygon will sometimes be a digon (a great circle plane intersecting only 2 vertices), as in the case of the the planes orthogonal to the 24-cell's hexagonal planes. The choice of polygons will (almost) determine the only possible range for the 𝜂 coordinate; the only remaining variable is the multiplicity 𝓂 of the coordinates. {| class="wikitable mw-collapsible {{{collapsestate|mw-expanded}}}" style="white-space:nowrap" !colspan=8|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Efn|name=Hopf coordinates key}} of the regular convex 4-polytopes |- ! ![[W:5-cell|5-cell]] ![[W:16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[W:24-cell|24-cell]] ![[W:600-cell|600-cell]] ![[120-cell]] |- !Digons |[[#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] | | | | |{{font color|green|(<small>{<2}𝜋, {<150}{{sfrac|𝜋|300}}, {<2}𝜋</small>)₁}} |- !Squares | |[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] | |rowspan=2|[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]]<BR>[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<15}{{sfrac|𝜋|30}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<75}{{sfrac|𝜋|150}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |- !Rectangles | | |[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] | | !Pentagons | | | | |[[#Great circle pentagons of the 600-cell|{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₅}}]] |{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₁}} |- !Hexagons | | | |[[#Great circle hexagons and squares of the 24-cell|{{font color|red|({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<5}{{sfrac|𝜋|10}}, {<4}{{sfrac|𝜋|2}}</small>)₁}} |[[#Great circle squares and hexagons of the 120-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤24}{{sfrac|𝜋|48}}, {<4}{{sfrac|𝜋|2}}</small>)₁}}]] |- ! | | | |[[#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<10}{{sfrac|𝜋|20}}, ({<6}{{sfrac|𝜋|3}}</small>)₃}} |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}},{<50}{{sfrac|𝜋|100}},{<6}{{sfrac|𝜋|3}}</small>)₃}} |- !Decagons | | | | |[[#Great circle decagons and hexagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤1}{{sfrac|𝜋|2}}, {<6}{{sfrac|𝜋|3}}</small>)₁}}]] |{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {<10}{{sfrac|𝜋|20}}, {<6}{{sfrac|𝜋|3}}</small>)₁}} | |- ! | | | | |[[#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |[[#Great circle decagons of the 120-cell|{{font color|red|(<small>{<10}{{sfrac|𝜋|5}},{≤5}{{sfrac|𝜋|10}},{<10}{{sfrac|𝜋|5}}</small>)₁}}]] |- !30-gons | | | | | |{{font color|green|(<small>{<30}{{sfrac|𝜋|15}},{≤15}{{sfrac|𝜋|30}},{<30}{{sfrac|𝜋|15}}</small>)₂₄}} |} ==== Great circle digons of the 5-cell==== One set of Cartesian origin-centered [[W:5-cell#Construction|coordinates for the 5-cell]] can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}} and radius {{radic|3.2}}: ( {{sfrac|4|{{radic|5}}}}, 0, 0, 0) (−{{sfrac|1|{{radic|5}}}}, 1, 1, 1) (−{{sfrac|1|{{radic|5}}}}, 1,−1,−1) (−{{sfrac|1|{{radic|5}}}},−1, 1,−1) (−{{sfrac|1|{{radic|5}}}},−1,−1, 1) Rescaled to unit radius and edge length {{sfrac|{{radic|5}}|{{radic|2}}}} these coordinates are: ( 1, 0, 0, 0) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) {| class="wikitable" !colspan=2|Great circle digons of the 5-cell<BR> Cartesian{{s|2}}({ , , , })<BR> Hopf{{s|2}}(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( 1, 0, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 1{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |(-{{sfrac|1|4}},-{{sfrac|{{radic|5}}|4}}, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|4}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|5}}|4}} ≈ 0.559}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 3{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 4{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} | |} ==== Great squares of the 16-cell ==== {| class="wikitable floatright" | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''xy'' plane |- |( 0, 1, 0, 0)||( 0, 0,-1, 0) |- |( 0, 0, 1, 0)||( 0,-1, 0, 0) |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''wz'' plane |- |( 1, 0, 0, 0)||( 0, 0, 0,-1) |- |( 0, 0, 0, 1)||(-1, 0, 0, 0) |} |}The 8 vertices of the 16-cell lie on the 4 coordinate axes and form 6 great squares in the 6 orthogonal central planes. The Cartesian axes lie on the diagonals of the square tables, which resemble the great squares. Rotate the tables 45 degrees clockwise for a vertex up orientation, and another 90 degrees for the standard ''xy'' orientation. By convention rotations are always specified in two completely orthogonal invariant planes xy (whose vertices are numbered counterclockwise by 𝜉_''xy'') and wz (whose vertices are numbered counterclockwise by 𝜉_''wz''). The rotation in the xy plane does not move points in the wz plane, and vice versa. In the 16-cell these two simple rotations rotate disjoint sets of 4 vertices each (because completely orthogonal planes intersect only at the origin and share no vertices). The 𝜂 coordinate of the 4 vertices in the xy plane is 0 and the 𝜂 coordinate of the 4 vertices in the wz plane is 1 ({{sfrac|𝜋|2}}). The w and z coordinates of the vertices in the xy plane are 0 regardless of the rotational position of the wz plane (the 𝜉_''wz'' coordinate), and the x and y coordinates of the vertices in the wz plane are 0 regardless of the rotational position of the xy plane (the 𝜉_''xy'' coordinate). {| class="wikitable" !colspan=2|Great squares of the 16-cell<BR> Cartesian{{s|2}}({0, ±1, 0, 0})<BR> Hopf{{s|2}}(<small>{<3}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<3}{{sfrac|𝜋|2}}</small>) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''xy'' plane |- !(𝜉_''xy'', 0, 0)||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 0, 1, 0, 0)||( 0, 0,-1, 0) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 1, 0)||( 0,-1, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}0: 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}0: 1}} |} | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''wz'' plane |- !(0, {{sfrac|𝜋|2}}, 𝜉_''wz'')||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 1, 0, 0, 0)||( 0, 0, 0,-1) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 0, 1)||(-1, 0, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 0}} |} |} ====Great rectangles (60 degree planes) of the 8-cell==== None of the 8-cell's 16 vertices lie in the 6 orthogonal central planes. The "north pole" is not a vertex, and 0 does not appear as a value in any (Hopf or Cartesian) coordinate. Each of the 8-cell's eight {{radic|4}} long diameters joining two antipodal vertices lies 45 degrees ({{sfrac|𝜋|4}}) off each of the 4 Cartesian coordinate axes. The Hopf 𝜼 coordinate is {{sfrac|𝜋|4}} for all the vertices, and the 𝜉_''xy'' and 𝜉_''wz'' coordinates are even and odd multiples of {{sfrac|𝜋|4}} respectively. Although the 16 vertices do not lie in the 6 orthogonal central planes, they do lie (by fours) in central planes, but the central polygons they form are rectangles (not squares), and the planes are inclined at 60 degrees ({{sfrac|𝜋|3}}) to each other and to the orthogonal central planes. These 16 great rectangles measure {{radic|1}} by {{radic|3}}, and their {{radic|1}} edges are opposite pairs of 8-cell edges. Their {{radic|3}} edges are interior chords of the 8-cell: long diagonals of the 8 cubic cells.{{Efn|In the 24-cell we find these 16 central rectangles inscribed in the 16 central hexagons. The 8-cell's great rectangles are the same central planes as the 24-cell's great hexagons, but the 8-cell has only 4 of the 6 hexagon vertices. The 16-point (8-cell) is a 24-point (24-cell) with 8 points (square pyramids) lopped off.}} Because there is only one 𝜼 coordinate value, only one table is required, but note that the table is not a great square but a duocylinder (its opposite edges are identified).{{Efn|For fixed {{mvar|η}} Hopf coordinates describe a torus parameterized by {{math|''ξ''_''xy''}} and {{math|''ξ''_''wz''}}, with {{math|''η'' {{=}} {{sfrac|π|4}}}} being the special case of the [[W:Clifford torus|Clifford torus]] in the {{mvar|xy}}- and {{mvar|wz}}-planes. All vertices of the 8-cell lie on the Clifford torus, a "flat" 2-dimensional surface embedded in the 3-sphere. The Clifford torus divides the 3-sphere into two congruent ''solid'' tori. [[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|In a stereographic projection]], the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized.}} Each row, column or diagonal in the table is a great rectangle. {| class="wikitable" ! colspan="2" |Great rectangles (60 degree planes) of the 8-cell Cartesian{{s|2}}(<small>±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}</small>)<BR> Hopf{{s|2}}({0 2 4 6}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}) |- | {| class="wikitable" !(𝜉_''xy'', {{sfrac|𝜋|4}}, 𝜉_''wz'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !0{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} |- !2{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !4{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%| 0}} !{{font|size=75%|−1}} |- !6{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%|−1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} ==== Great squares and rectangles of the 24-cell ==== The great square coordinates of the 16-cell (above), combined with the great rectangle coordinates of the 8-cell (above), comprise a set of coordinates for the 24-cell. Because the 16-cell lies vertex-up in this coordinate system, so does the 24-cell. ==== Great squares of the 24-cell ==== Another useful set of coordinates for the 24-cell is comprised solely of orthogonal great squares. In this coordinate system the 24-cell lies cell-up, and the great squares are aligned with the squares of the coordinate lattice. {| class="wikitable" !Great squares of the 24-cell<BR>Cartesian{{s|2}}(<small>±{{radic|{{sfrac|1|2}}}}, ±{{radic|{{sfrac|1|2}}}}, 0, 0</small>) |- | {| class="wikitable" !Hopf{{s|2}}({<4}{{sfrac|𝜋|2}}, {{sfrac|𝜋|4}}, {<4}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" !(𝜉_''i'', {{sfrac|𝜋|4}}, 𝜉_''j'') !0{{sfrac|𝜋|2}} !1{{sfrac|𝜋|2}} !2{{sfrac|𝜋|2}} !3{{sfrac|𝜋|2}} !sin !cos |- !0{{sfrac|𝜋|2}} |(<small> {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%| 1}} |- !1{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !2{{sfrac|𝜋|2}} |(<small>-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}},-0, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}},-0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%|-1}} |- !3{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} |- | {| class="wikitable" !Hopf{{s|2}}({1 3 5 7}{{sfrac|𝜋|4}}, {≤1}{{sfrac|𝜋|2}}, {1 3 5 7}{{sfrac|𝜋|4}})₄ |- | {| class="wikitable" !(𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" !(𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} |} |} ==== Great squares and rectangles of the compound of dual 24-cells ==== Two sets of coordinates for the 24-cell have now been given (above). In the first (great squares and rectangles) the 24-cell lies vertex-up, and in the second (great squares only) it lies cell-up. The 24-cell being a self-dual 4-polytope, these two 24-cells are duals of each other, the vertices of one lying at the cell centers of the other, and the union of their two sets of coordinates is a 48-vertex compound of duals of the same radius. In this compound of two 24-cells, 24 16-cells are inscribed: the 3 inscribed in each of the dual 24-cells, and 18 others which span the two 24-cells.{{Sfn|Waegell|Aravind|2009}} Each of these 24 16-cells, with its 4 orthogonal axes and 6 orthogonal planes, constitutes an equivalent ''basis'' for a Cartesian coordinate system, and contains three pairs of completely orthogonal planes, each pair of which intersects all 8 of the 16-cell's vertices. The compound of two 24-cells has 24 axes and 24 bases, with each basis consisting of four axes and each axis occurring in four bases. The 3 16-cells inscribed in each 24-cell are disjoint from each other and the dual 24-cell's 3 inscribed 16-cells. Each 24-cell has 18 central squares, and the 18 spanning 16-cells are each comprised of 4 vertices comprising a great square from one of the 24-cells, and another 4 vertices comprising a great square from the dual 24-cell.{{Efn|In each of the 18 16-cells, the two central squares from dual 24-cells are orthogonal, but not completely orthogonal. Each is already completely orthogonal to another central square within the same 24-cell (within the same 16-cell), and in 4-space a plane cannot be completely orthogonal to more than one other plane through the same point. Although their two sets of 4 vertices are disjoint, that is not because their square planes are completely orthogonal; rather, their planes intersect in a line, but their vertices remain disjoint because the line of intersection does not pass through any of their vertices.}} ==== Great hexagons and squares of the compound of dual 24-cells ==== In a single 24-cell, the hexagonal central planes lie at 60 degrees to each other and to the square central planes. The central planes orthogonal to the hexagonal planes are digons: they intersect only 2 vertices. Consequently it is not possible to find Hopf coordinates for the 24-cell in which both the 𝜉_''i'' and 𝜉_''j'' orthogonal invariant planes contain hexagons. However, the 24-cell and its unscaled dual form a compound in which the dual 24-cells are separated by a Clifford displacement (an isoclinic rotation) of 30 degrees. The hexagonal planes are still not orthogonal to each other in this compound (they are inclined at 30 degrees or 60 degrees to each other), so it is still impossible to find a 6 x 6 array of Hopf coordinates, but the hexagonal planes of one 24-cell are orthogonal to the square planes of the other. In this compound of 48 vertices, a hexagonal wz plane and a square xy plane can be the invariant planes of a 6 x 4 array of Hopf coordinates. {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the compound of dual 24-cells Cartesian{{s|3}}(<small>±_, ±_, _, _</small>)<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {1 2}{{sfrac|𝜋|6}}, {<small><4</small>}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" style="white-space:nowrap;" !|(𝜉_''i'', 1{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|2}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|1|2}}}} |} |} ==== Great hexagons and digons of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and digons of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜋, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} -1}} |} |} ==== Great hexagons and squares of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 0𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( 0, 1, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |( 0, {{sfrac|1|2}}, 0, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{sfrac|3|4}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{sfrac|3|4}}, 0) |( 0, 0, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( 1, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |( {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 1𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}|| {{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( a, , 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} -1}} |} |} ==== Dual fibrations ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[fiber bundle]] of Clifford parallel great circles is a discrete [[Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once. {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell<BR> Hopf ({<<small>6</small>}{{sfrac|𝜋|3}}, {<small><10</small>}{{sfrac|𝜋|20}}, {<small><2</small>}𝜋)₁<BR> Cartesian ({<small>0, ±1, 0, 0</small>}) (<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>) ([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle pentagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle pentagons of the 600-cell<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>])<BR> Hopf{{s|3}}({<small>0 2 4 6 8</small>}{{sfrac|𝜋|5}}, {<small><24</small>}{{sfrac|𝜋|48}}), {<small>1 3 5 7 9</small>}{{sfrac|𝜋|5}})₅ |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'',{{sfrac|𝜋|4}}, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle squares and hexagons of the 120-cell==== {| class="wikitable" !colspan=1|Great circles of the 120-cell:<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {<small>≤24</small>}{{sfrac|𝜋|48}}, {<small><4</small>}{{sfrac|𝜋|2}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]){{s|3}}... |- | |} ====Great circle decagons and hexagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell:<BR> Hopf{{s|3}}({<10}{{sfrac|𝜋|5}}, {<small>≤1</small>}{{sfrac|𝜋|2}}, {<<small>6</small>}{{sfrac|𝜋|3}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle decagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons of the 600-cell:<BR> Hopf{{s|3}}({<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}}, {<small>0 1 2 3 4 5</small>}{{sfrac|𝜋|10}}, {<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}})₅<BR> Cartesian{{s|3}}... |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 0{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0))||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !style="white-space:nowrap;"|{{font|size=75%|sin}}||{{font|size=75%|𝜂 {{=}} 0}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !style="white-space:nowrap;"|{{font|size=75%|cos}}||{{font|size=75%|𝜂 {{=}} 1}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 1{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}},{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, 0)||(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}},-{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}},-{{sfrac|1|4}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ^-1|2}} ≈ 0.309}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} b ≈ 0.951}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} {{s|5}}<small>{{sfrac|bϕ|2}} ≈ 0.769{{s|5}}ab ≈ 0.559{{s|5}}{{sfrac|bϕ^-1|2}} ≈ 0.294{{s|5}}b² ≈ 0.905</small> |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 2{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 3{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 4{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 5{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |} == Equilateral rings == Equilateral rings are those which can be constructed out of equilateral triangles on the circumference of a sphere. === Borromean equilateral rings === The {1,1,1} torus knot. The vertices of the regular icosahedron form five sets of three concentric, mutually [[orthogonal]] [[golden rectangle]]s, whose edges form [[Borromean rings]]. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures <math>2\times 4</math>. These three rectangles are the shortest possible representation of the Borromean rings using only edges of the [[integer lattice]]. ... == Kinematics == In 3D we have the kinematic transformations of the cuboctahedron (cuboctahedron, icosahedron, jessen's, golden icosa?, octahedron-2, tetrahedron-4?) and their duals, the transformations of the dodecahedron: two sets of nesting Russian dolls (or perhaps one set?). In 4D we apparently have instances of the cuboctahedron nestings in the 600-cell (and perhaps the dodecahedron nestings as well, in the 120-cell?). This suggests that the unit-radius sequence of 4-polytopes may contain dynamic as well as static nestings. From [[W:Kinematics of the cuboctahedron#Duality of the rigid-edge and elastic-edge transformations|Kinematics of the cuboctahedron § Duality of the rigid-edge and elastic-edge transformations]]: <blockquote> Finally, both transformations are pure abstractions, the two limit cases of an infinite family of cuboctahedron transformations in which there are two elasticity parameters and no requirement that one of them be 0. ... In engineering practice, only a tiny amount of elasticity is required to allow a significant degree of motion, so most tensegrity structures are constructed to be "drum-tight" using nearly inelastic struts ''and'' cables. A '''tensegrity icosahedron transformation''' is a kinematic cuboctahedron transformation with reciprocal small elasticity parameters.</blockquote> From [[W:24-cell#Double rotations|24-cell § Double rotations]]: <blockquote>The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes of rotation at once.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. </blockquote> This suggests that the reciprocal limit-case kinematic transformations (with one orthogonal elasticity parameter or the other equal to 0) may be expressable as double rotations, considering the relativity of such transformations.{{Efn|name=transformations}} === Completely orthogonal planes === In three dimensions (on polyhedra) there are no disjoint great circles. Every pair of great circles intersects at two points, the endpoints of a diameter of the sphere. But in four dimensions (on polychora) every great circle is disjoint from exactly one other great circle: the one to which it is completely orthogonal. If we consider the two polyhedral great circles' common diameter to be a common axis of rotation, we can see that rotating either circle about that axis generates the whole polyhedron; thus either circle by itself can generate the whole polyhedron by rotation. But in four dimensions two completely orthogonal great circles have no common axis of rotation (no points at all in common, all their points are disjoint). Clearly either circle by itself cannot generate the entire polychoron by rotation about a fixed axis. Rotating each circle about an axis generates only half the points on the 3-sphere - rotating the other circle generates the other half of them. Rotation about a fixed axis in four dimensions necessarily leaves an entire plane fixed, and generates only a 3-dimensional polyhedron. If the great circle in the xy plane is rotated about the y axis, only a 2-sphere is generated, and all the points on the 3-sphere outside the hyperplane w = 0 will be left out. In the 24-cell and the 8-cell, which are radially equilateral, ... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. {{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Coxeter mirrors ==== [[W:Coxeter group|Coxeter group]] theory (the mathematics of [[W:Polytopes|polytopes]] of any number of dimensions) can be informally described as a ''finite closed system of mirrors'', or the [[W:Geometry|geometry]] of multiple [[W:Mirror#Mirror images|mirror images]]. Mathematically it is the equivalent of the theory of finite [[W:Reflection group|reflection groups]] and [[W:Root system|root systems]], expressed in a different mathematical language. But unlike those [[W:Group theory|group theory]] languages, its principal objects can be defined in the most intuitive and elementary way: <blockquote> Imagine a few (semi-transparent) mirrors in ordinary three dimensional space. Mirrors (more precisely, their images) multiply by reflecting in each other, like in a [[W:Kaleidoscope|kaleidoscope]] or a [[W:Gallery of mirrors|gallery of mirrors]]. A ''closed system of mirrors'' is what we see when we look into such a kaleidoscope.{{Sfn|Borovik|2006|loc=§1. Mirrors and Reflections|pp=18-19}}</blockquote> Coxeter refers to the space between two parallel mirrors as "the region of possible objects". This suggests that space itself is being generated by the objects that it contains, which parallel reflections multiply infinitely. When the mirrors are not parallel, the multiplication may be finite rather than infinite (provided the dihedral angle between the mirrors is a submultiple of 𝝅). The space between two or more such ''intersecting'' mirrors is called the fundamental region, and it constitutes a proto-space of a finite number of dimensions with mirrors as its bounding walls. Coxeter says "the lines of symmetry or circles of symmetry or planes of symmetry are mirrors reflecting the whole pattern into itself. We count these circles of symmetry by counting their pairs of antipodal points of intersection with a single equator."{{Sfn|Coxeter|1938}} He characterizes great circles of symmetry in terms of the Petrie ''h''-gon i.e. the 3''h''/2 circles of symmetry possessed by each Platonic solid. For example the Petrie polygon of the octahedron and the cube is the hexagon, so they have 18/2 = 9 great circle planes of symmetry (mirrors). Each is generated by placing a single point-object in the fundamental region (off the surface of any mirrors); reflected in the mirrors it multiplies into all the vertices of the polytope. ==== Translation-rotations ==== An object displacement in space may be a rotation (which leaves at least one point invariant), a translation (which does not), or a combination of both.{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations|pp=33-38}} The circular path of a rotation may be combined with a translation in the axial direction of the rotation yielding a ''screw-displacement'', the general case of a displacement.{{Sfn|Coxeter|1973|loc=§3.14|ps=; "''Every displacement is a screw-displacement'' (including, in particular, a rotation or a translation)."|p=38}} In three dimensions, a screw-displacement is a simple helix, as its name suggests. In four dimensions, the circular path of an isoclinic rotation is already a helix (a geodesic isocline), and there are four orthogonal axial directions of rotation.{{Sfn|Coxeter|1973|loc=§12.1 Orthogonal transformations|pp=213-217|ps=; "The general displacement preserving the origin in four dimensions is a ''double rotation''.... The two completely orthogonal planes of rotation are uniquely determined except when 𝜉₂ {{=}} 𝜉₁, in which case... we have a ''Clifford displacement''."}} In the unit-radius 24-cell an isoclinic rotation by 60° moves each vertex {{radic|3/4}} ≈ 0.866 in each of four orthogonal directions at once, a total Pythagorean distance as if it had moved straight along a combined {{radic|3}} ≈ 1.732 chord. When the rotation is combined with a unit translation, the {{radic|1}} translation vector must be divided among all four rotation vectors. The vertex moves {{radic|3/4 + 1/4}} = {{radic|1}} in each of the four orthogonal directions, moving a combined Pythagorean distance of {{radic|4}}, the maximum ''unit displacement'' in 4 dimensions: the distance which is the [[W:Tesseract#Radial equilateral symmetry|long diameter of the 4-hypercube (tesseract)]]. This movement ''could'' take the vertex to its antipodal vertex {{radic|4}} away, if the direction of the translation is so aligned, but in all other cases it will take it to a point outside the 24-cell. ==== Relative screw displacement ==== (screw displacement) QT = QR² = R⁴ = Q² (double rotation){{Sfn|Coxeter|1973|P=217|loc=§12.2 Congruent transformations}} A screw displacement in four dimensions is equivalent to a double rotation, by the principle of relativity. There are only two kinds of screw displacements possible in only ''four'' dimensions: the single kind (which also occurs in three dimensions), and the double kind (which requires four dimensions). The latter kind of screw displacement is inherently double, the product of 4 reflections, just like a double rotation. In fact it ''is'' just a double rotation, seen from a moving inertial reference frame. A product of two reflections is a (simple) rotation, unless the reflecting facets are exactly parallel, in which case it is a translation. In other words, a translation is just a rotation on a circle of infinitely long radius (a straight line). A screw displacement is just a double rotation in which one rotation (the one which is the translation) has infinitely long radius, i.e. a vanishingly small angle of rotation (near 0 degrees) compared to the angle of the other rotation (between 0 and 90 degrees). The screw displacement looks like a simple rotation within a three dimensional reference frame that is in uniform translation on the 4th dimension axis at near-infinite velocity. In the case of actual moving objects, no actual translation is an infinite straight line, and no velocity is infinite; any moving object that describes a screw-displacement is presumably moving on a curved translation under the influence, at least, of some distant gravitational force, however miniscule, and the radius of the translation-rotation it describes is merely very long, not infinite. So we can say that there can be only one range of situations in actuality, no perfectly straight translations but only double rotations of more or less eccentricity, which will appear to be simple rotations inside a 3-dimensional reference frame moving uniformly along the 4th dimension "translation" axis. Its eccentricity is merely a matter of choice of reference frame: it looks like a simple rotation in a reference frame moving uniformly with the translation, and a double rotation (of perhaps extreme eccentricity) in a reference frame that is not moving with the translation. Within this range of possibilities, only one possibility is ''not'' eccentric: the case of the equi-angled double rotation, called an isoclinic rotation or Clifford displacement. Since the ratio of eccentricity is a matter of choice of reference frame, we may adopt as our preferred reference frame (of any actual screw displacement occuring in practice) the inertial reference frame in which the double rotation is isoclinic. ==== Total internal reflection ==== The phenomenon in physics known as [[W:Total internal reflection|total internal reflection]] keeps light confined within one strand of a [[W:Optical fiber|fiber optic cable]]. Isoclinic rotations and screw displacements in 4-dimensional space are both the consequence of four symmetrical reflections, and their propagation corresponds to a total internal reflection within the 3-sphere. Consequently 4-dimensional space itself acts as a [[W:waveguide|waveguide]] for isoclinically rotating objects during their translation. This provides a purely geometric model for the [[w:inertia|inertia]] of mass-carrying objects, and for light-wave propagation. ==== Isoclines ==== In an isoclinic rotation the vertices of a 4-polytope such as the 24-cell move on ''isoclines'', which are helical circles that wind through all four dimensions. Isoclinic rotations are [[W:chiral|chiral]], occuring in left-handed and right-handed mirror-image pairs in which the moving vertices reach different destinations along left or right paths. The isoclines themselves however (the helical paths of the moving vertices) are not chiral objects: they are non-twisted (directly congruent) ''circles'', of a special 4-dimensional kind. Every left isocline path in a left-handed rotation acts also as a right isocline path in some right-handed rotation, in some ''other'' left-right pair of isoclinic rotations. Isoclinic rotations and their isoclines occur as fibrations (fiber bundles of non-intersecting but interlinked circles), with each fibration consisting of a single distinct left-right pair of isoclinic rotations. Each distinct left (or right) rotation has some number of isoclines, which are the circular paths along which its vertices orbit, with each vertex confined to a single isocline circle throughout the rotation. The multiple isoclines of a distinct left or right rotation do not intersect each other; they are Clifford parallel, which means that they are curved lines which are parallel to each other, in the sense that they are the same distance apart at all of their corresponding (nearest) points. Thus the moving vertices in an isoclinic rotation circulate in parallel disjoint sets. In the 24-cell's characteristic kind of isoclinic rotation, the moving vertices circulate on skew hexagon isoclines, in 4 parallel disjoint sets of 6 moving vertices each. This characteristic kind of isoclinic rotation occurs in four different fibrations: there are four distinct left-right rotation pairs. In each distinct left (or right) rotation, there are 4 Clifford parallel isoclines, each of which is a helical circle through 6 vertex positions. The 4 disjoint circles of 6 vertices pass through all 24 vertices of the 24-cell, just once. Although the four isocline circles do not intersect, they do pass through each other as do the links of a chain, but unlike linked circles in three-dimensional space, they all share the same center point. ==== Polygrams and cell rings ==== The isoclines of [[W:24-cell#Isoclinic rotations|24-cell isoclinic rotations]] in ''hexagonal'' central planes have 6 chords which form a [[W:Skew polygon|skew]] [[W:hexagram|hexagram]]. Every [[W:24-cell#Helical hexagrams and their isoclines|hexagram isocline]] is contained within the volume of a distinct [[W:24-cell#Cell rings|ring of 6 face-bonded octahedral cells]] which, like its axial great circle hexa''gon'' is an equatorial [[W:24-cell#Rings|ring of the 24-cell]]. Each 6-cell ring contains the left and right isoclines of a distinct left-right pair of isoclinic rotations. [[W:24-cell#6-cell rings|The 6-cell ring itself]] is not a chiral object because it contains ''both'' mirror-image isoclines: they are twisted in opposite directions (around each other), but the 6-cell ring that contains them both has no [[W:Torsion of a curve|torsion]]. ==== Reflections ==== Because each octahedral cell volume can be subdivided into 48 orthoschemes (the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the octahedron]] and [[W:24-cell#Characteristic orthoscheme|of the 24-cell]]), we can be more precise in describing the cell ring each isocline stays within. Within the 6-cell ring of face-bonded spherical octahedra is a ring of face-bonded spherical characteristic tetrahedra that contains the isocline. Several characteristics of this ring are evident from the nature of [[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoschemes]]: it consists of alternating left-hand and right-hand orthoschemes (mirror images of each other in their common face), and each of them contains one and only one vertex of the 24-cell.{{Efn|In the 24-cell 48 [[W:24-cell#Characteristic orthoscheme|characteristic 4-orthoschemes (5-cells)]] meet at each of the 24 vertices, and 48 [[W:Octahedron#Characteristic orthoscheme|characteristic 3-orthoschemes (tetrahedra)]] fill each of the 24 octahedral cells. The number of characteristic orthoschemes in a regular polytope is the ''order'' of its [[W:Coxeter group|symmetry group]], which for the octahedron is 48, and for the 24-cell is 1152 {{=}} 24 * 48.}} The sequence of orthoscheme vertices is the same as [[W:24-cell#Helical hexagrams and their isoclines|the sequence of isocline vertices]], except that it includes the vertices near-missed by the isocline as well as those the isocline intersects (12 distinct vertices instead of 6), and the orthoscheme sequence will have many more than 12 elements because it includes more than two orthoschemes incident to each vertex.{{Efn|The ring contains more orthoschemes than necessary to contain the isocline, because of our stipulation that all pairs of adjacent orthoschemes be face-bonded. The isocline intersects only a vertex, or only an edge, of some orthoschemes in the face-bonded ring. At each vertex it hits, the isocline passes between two orthoschemes that touch only at that vertex. Near each vertex that it misses, the isocline passes through an edge between two orthoschemes that touch only at that edge.}} === Physical space === We attempt to be more precise about the shape of this 4-space, and in particular, the cause of its shape, i.e. the relationship between the fundamental forces observed in nature and this spatial geometry. As Einstein did in his 1923 thought experiment, we identify the observed 3-dimensional cosmos (everything in it up to some large scale such as a galaxy) as a thin manifold embedded in a Euclidean (i.e. flat) 4-dimensional space of the kind elucidated by Coxeter. Further we postulate that every mass-carrying particle in this space is in motion at speed <math>c</math> relative to the 4-dimensional space itself. The 4-space therefore has a quasi-ether-like existence insofar as it defines a field at absolute rest, relative to which the motion of all particles at speed <math>c</math> can be universally compared, with the important provision that no particle, anywhere, is ever at absolute rest itself with respect to this field. The condition of absolute rest is an abstract condition attributable only to the field, and never to any tangible object. Thus the field itself (4-dimensional Euclidean space) is an abstraction somewhat more tangible than Mach's relative space, but much less tangible than the luminous ether, much as Einstein found 4-dimensional spacetime to be. Directions and distances can be fixed universally within the Euclidean 4-space field (they are invariant for all observers regardless of their direction of motion within the field), but locations can only be relative to some object (not to the field itself), and all 4-dimensional velocities are invariant: they are always <math>c</math> with respect to the field, for any mass-bearing particle or observer. Einstein's general relativity identifies gravity as a fictitious force, attributable to the shape of the 3-dimensional manifold rather than to an attractive force acting instantaneously at a distance. The 3-dimensional manifold is said to be singular and universal (all objects in the universe lie within it), but its shape varies by location. It is assumed to curve or dimple in the vicinity of massive objects, such that other objects fall into the dimples naturally in the course of following straight-line paths (geodesics) through it. In general relativity 3-dimensional space is flat near each observer, but there is no universally flat space except in regions far removed from massive objects, i.e. in places where the simplifications of the theory of special relativity can be assumed. But in Euclidean relativity this 3-dimensional manifold is embedded in a 4-dimensional Euclidean space, and that 4-space field is flat universally, at all times for all observers. Furthermore, we only assume that the 4-dimensional space is singular and universal; there might be more than one 3-dimensional manifold embedded within it, and the 3-manifolds do not necessarily intersect. In Euclidean relativity we expect that not just gravity, but all the fundamental forces observed in nature, are an expression of the local geometry of a 3-space manifold embedded in Euclidean 4-space. By ''expression'' we mean the consequence of a transformation such as a reflection, rotation or translation, i.e. operations of the fundamental Coxeter symmetry groups, which characterize the Euclidean geometry of the universal space in which the 3-space manifolds are embedded. === Closed 3-manifolds embedded in 4-space === The only reason to suppose there is only one such closed, curved soap-bubble 3-manifold in our 4-space universe is the assumption that every particle in the universe had a common origin, at a single point in 4-space and a single moment in time in a big bang, and even in that case there could be many such soap-bubble 3-manifolds in existence now. One can certainly model the observed universe as a single closed, curved 3-manifold, and cosmologists do, but there is no more proof that this model is the correct one than there was for the model with the earth at the center of the universe. Whether we determine that light propagates through 4-space in straight lines, or only on geodesic curves along 3-manifolds, we can only determine ''by looking'' that the space ''near'' us is resolutely 3-dimensional (not admitting the construction of four mutually perpendicular axes, only three). When we look out very far, at distant galaxy clusters for example, we have no way of determining whether we are looking through three dimensional space or four dimensional space. All those distant objects we see ''might'' lie in the same 3-manifold (perhaps on the same rough 3-sphere) that we do, but why should they have to? Might they not lie on separate 3-sphere soap bubbles, vastly distant from ours, whether or not all the soap bubbles had a common origin at one place and time? When we consider the ways in which particles propagating at the speed of light might reach us, considering that we ourselves are formations of particles propagating at the speed of light (all together in almost the same direction), it is clear that we ought not to expect to be overtaken by such particles emanating from stars on the opposite side of our own 3-manifold (from our antipodes, so to speak), because even such a particle redirected exactly backwards along the proper time axis of a star at our antipodes could only follow us along our opposite-direction proper-time axis at a fixed distance forever, never overtaking us, as we travelled in the same direction through 4-space at the same speed <math>c</math> forever. Or, since our own path through 4-space is a helical one (as we are engaged in numerous concentric orbits), if the pursuing particle's path through 4-space were a straighter one, it might in principle overtake us eventually, but probably not in our actual experience, and never as a particle moving relative to us at nearly the speed <math>c</math>. Therefore we should not expect to receive such particle radiation from the backside stars of our own 3-manifold. === The speed of light === So far, however, these considerations can apply only to mass particle radiation, not to light signals, since we have not yet described how light particles (photons) propagate through 4-space. We have suggested that elementary rigid objects propagate themselves by discrete Coxeter transformations,{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} </blockquote>|name=transformations}} that atomic mass particles are elementary rigid objects in some sense, and that all such particles are transforming at a constant rate <math>c</math> in various directions through Euclidean 4-space. But so far, the motion of mass particles is the only kind of motion we have described; we have not given an account of the nature of light signals, or the manner of their propagation, except to observe that light signals propagate through 4-space ''faster'' than <math>c</math>, as a consequence of the fact that they are observed to propagate through ''3-space'' at speed <math>c</math>. How much faster than <math>c</math> must photons be traveling through 4-space, since they appear to be traveling at <math>c</math> relative to two observers: one in the reference frame of the electron emitting the photon, and one in the reference frame of the electron absorbing the photon? If both observers are themselves traveling through 4-space at speed <math>c</math> as we have stipulated, then even in the case where their direction through 4-space is the same (they are at rest with respect to each other in the same reference frame), a photon that passes between them must travel at speed <math>\sqrt{2}c</math> if it makes the trip in a straight line, or even faster if it zig-zags in some fashion. Fortunately, this requirement is not at all paradoxical, since in a system of particles translating themselves through 4-space at the rate of <math>c</math> transforms per unit time, some things ''do'' move faster than speed <math>c</math>. A transforming rigid object with a translational motion at rate <math>c</math> may simultaneously have an orthogonal rotational motion at rate <math>c</math>, such that its component parts (e.g. each vertex of a rotating-translating 4-polytope) may displace themselves in 4-space ''more'' than one object-diameter in each discrete transformation; the combined rotating-and-translating velocity through 4-space of a ''component'' may be as much as twice the translational velocity of the whole rigid object, <math>\sqrt{4}c</math> rather than <math>\sqrt{1}c</math>, the [[W:8-cell#Radial equilateral symmetry|diagonal of an atomic unit 4-cube]] rather than its edge length. But the component points of such a rotating rigid object are all traveling in different directions at any instant, and the combined motion of the object as a whole cannot be other than <math>c</math>. Therefore a propagating light signal (a photon) is not a rigid atomic object, but some propagation of one of its component parts. Of course this agrees perfectly well with our understanding of photons as emissions of electrons, even if electrons are themselves rigid atomic objects translating themselves through 4-space at the rate of <math>c</math> transforms per unit time. The only "paradox" is linguistic in nature: <math>c</math> is not the "speed of light", it is the speed of matter (all mass-carrying particles) through 4-space. The actual ''speed of photons'' through 4-space is <math>2c</math>, as opposed to their observed speed through 3-space of <math>c</math>. === ... === A light signal (photon) propagates at speed c relative to either the emitting or the absorbing inertial reference frame (which reference frames are themselves in motion at speed c relative to their common 4-space stationary reference frame, but perhaps in different directions through that 4-space). ==== How soap bubble 3-manifolds behave in 4-space ==== 120 similar 2-sphere soap bubbles (spherical dodecahedra) tiling the 3-sphere, meeting at 120 degrees three-around each edge, and four at each vertex.{{Sfn|Stillwell|2001|p=24|loc=Figure 7. Soap bubble 120-cell}} === Atmospheric 3-membrane === What if the 3-space we observe (the visible universe) were filled with a gas, contained in some manner within the (thin) 3-membrane? Like the (thin) atmosphere of the earth (in a 3-dimensional analogy). In fact it is gas-filled, if the 4-space inside and outside the 3-membrane is empty. That difference is precisely what defines the 3-membrane: it is the 4-space which is not empty. It is not continuously full, as it is a cloud of discrete particles like a gas, and the density of particles is very low in most places, but within the 3-membrane it is never zero. The 3-membrane(s) is the surface(s) of 4-polytope(s) with a very large number of vertices (a function of the number of atoms in the universe, if those vertices are enumerated in some manner that includes the universe's plasma matter), and as in any 4-polytope all its elements lie on a 3-dimensional surface (albeit in the case of plasma on a rather inconsequential and rapidly changing 3D surface). === Configurations in the 24-cell === [[File:Reye configuration.svg|thumb|Reye's configuration 12₄16₃ of 12 lines (3 orthogonal groups of 4) intersecting at 16 points.]] The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye’s configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} In its most elemental expression, Reye's configuration is a set of 12 lines which intersect at 16 points, forming two disjoint cubes.{{Efn|The basic expression of Reye's configuration 12₄16₃ of 12 lines and 16 points occurs 12 times in the 24-cell, as the 16 vertices of two opposite (completely orthogonal) cubical cells (in one of the 3 inscribed 8-cells), and the 12 geodesic straight lines (hexagonal great circles) on which the cubes' parallel edges lie (orthogonally in the 3 dimensions of 3-sphere space, embedded in a Euclidean space of 4 orthogonal dimensions).}} It has multiple expressions in the 24-cell.{{Efn|An expression of Reye's configuration 12₄16₃ with respect to isoclinic rotations of the rigid 24-cell: 12 points are reached by 4 half hexagram isoclines in each left ''or'' right 360 degree isoclinic rotation characteristic of the 24-cell, and each of the 16 half hexagram isoclines in the left ''and'' right 360 degree rotations reaches 3 points.}} If the proton and neutron are indeed the hybrid fibrations we have identified, on that assumption we can count the number of valid protons and neutrons that can coexist in the same 24-cell at the same time without colliding, and the total number of distinct configurations of multiple nucleons (the number of distinct nuclides) that a single 24-cell can contain. This will not limit the number of distinct nuclides which can exist, because the 24-cell can be compounded in four-dimensional space in several different ways, but it will quantify the number of nuclides which can occupy the first nuclear shell. We might ask of any less minimal configuration of rotations in a single 24-cell whether it in fact corresponds to an integral number of nucleons occupying the same nuclear shell. A valid ''maximal'' hybrid configuration of rotations in the 24-cell would have the largest number of moving vertices possible without collisions (perhaps 24).{{Efn|The valid maximal configurations do have 24 moving vertices. They include a configuration with 12 vertices rotating ''within'' each of the three 16-cells on 4 octagram isoclines of the same chirality (while remaining within 4 Clifford parallel moving square planes), and visiting all 8 vertices of each 16-cell in each double revolution. The same maximal configuration also has 12 vertices rotating ''among'' the three 16-cells on 4 hexagram isoclines (two left-right pairs) while remaining within 4 Clifford parallel moving hexagonal planes, and visiting all 24 vertices of the 24-cell in each double revolution.}} ====...==== If the particle energies are to be described as the angular momentum of isoclinic rotations of some kind, it is noteworthy that square isoclinic rotations will describe 16-cells, but not (by themselves) the 24-cell. Square isoclinic rotations are associated with the ''internal'' geometry of 16-cells: they are the [[W:16-cell#Rotations|characteristic rotations of the 16-cell]]. The chords of the single [[W:16-cell#Helical construction|isocline of a square isoclinic rotation]] (left or right) are the four orthogonal axes of the 16-cell (enumerated twice), and the isocline is a helical circle passing through all 8 vertices of the 16-cell. A left-right pair of square rotations covers all the elements of the 16-cell (including its 18 great squares and its 16 tetrahedral cells). Thus the square isoclinic rotations say all there is to say about the internal geometry of an individual 16-cell, but they have nothing at all to say about how three 16-cells combine to form a 24-cell. The hexagonal isoclinic rotations, the [[W:24-cell#Isoclinic rotations|characteristic rotations of the 24-cell]], do describe the whole 24-cell. Their hexagram isoclines wind through all three 16-cells, their {{radic|3}} chords connecting the corresponding vertices of pairs of disjoint 16-cells. If vertices moving in hexagonal isoclinic rotations are what carries the energy binding the three quarks together (at least in the case of the neutron), each neutron would require at least two vertices moving in hexagonal isoclinic orbits, for the following reason. We attributed electric charge to the chirality of isospin generally, so we may expect hexagonal isoclinic rotations to contribute to the nucleon's total electric charge, even though they are not intrinsic parts of the three quarks, but rather parts of the whole 24-cell. Since the hexagonal orbits span the three 16-cells equally, the contributions of their moving vertices must be of neutral charge overall. Therefore the neutron must possess pairs of left and right hexagonal isoclinic rotations: minimally, one vertex moving on a left hexagram isocline, and one moving on a right hexagram isocline, which cancel each other because they have exactly opposite isospin. They must be a left and a right isocline from the same fiber bundle, corresponding to the left and right rotations of the same set of Clifford parallel invariant hexagonal planes of rotation. Such a pair is a valid kinematic rotation, because left and right hexagram isoclines of the same fibration do not intersect, and so can never collide.{{Efn|Most, but not all, left and right pairs of isoclinic rotations have isocline pairs which are Clifford parallel and visit disjoint vertex sets. The exception is left and right square isoclinic rotations. Their left and right [[W:16-cell#Helical construction|octagram isoclines]] ''do'' intersect, and they each visit the same set of 8 vertices.}} ====...==== The chiral pair of hexagonal rotations combined with the various square rotations will be a valid hybrid rotation provided no vertex in a hexagram orbit ever collides with a vertex in an octagram orbit. The octagram orbits visit all 8 vertices of the 16-cell in which they are confined. In an up quark with the minimum two moving vertices there are 6 empty vertices at any moment in time, leaving room for cross-traffic. The moving vertex on a hexagram isocline (of which minimally there will be two, a left and a right) will intersect each 16-cell in two places (not antipodal vertices) at different times. If the two moving vertices are antipodal (a moving axis), they will intersect each 16-cell in two axes at different times. There are valid configurations of this set of ''minimal'' hybrid rotations. In up-down-up proton configurations, the 7 moving vertices may be chosen in various ways that avoid collisions. Similarly in down-up-down neutron configurations, there are 6 moving vertices and various valid configurations. These minimal configurations could be are protons and neutrons, or they could be fractional parts of whole nucleons, less minimal configurations of the same kind of hybrid rotation with more moving vertices. These minimal hybrid rotations fall short of the full fibration symmetry of an ordinary isoclinic rotation in various ways. The hexagonally-rotating vertices visit only 12 of the 24 vertices once.{{Efn|Each hexagonal fibration has a right isoclinic rotation on 4 Clifford parallel right hexagram isoclines, and a corresponding left rotation on 4 Clifford parallel left hexagram isoclines, in the same set of 4 Clifford parallel invariant hexagonal planes. The right and left rotations reach disjoint sets of 12 vertices.}} The two square-rotating vertices in each up 16-cell visit all 8 vertices twice; the one in each down 16-cell visits all 8 vertices once. The two hexagram vertices each rotate through three 16-cells, so even with the best synchronization, there will be an oscillation in the total number of moving vertices in each 16-cell at any one time. In these minimal configurations, the 16-cells and the 24-cell would be strangely unbalanced objects. More generally, any such balance would require solutions to the n-body problem for 7 and 6 bodies, respectively. We could attempt to remedy these deficiencies by adding more moving vertices on more isoclines, seeking to make nucleons which are ''hybrid fibrations'' with at least one moving vertex on each isocline of each kind of fibration. This more balanced configuration with complete fibrations can be achieved with 6 vertices moving on hexagram isoclines, synchronized so that there are two of these moving vertices in each 16-cell at once. The proton (or neutron) will be a valid hybrid fibration of the 24-cell if it is synchronized to avoid collisions, with 6 vertices moving over all 24 vertices on isoclines of hexagonal fibrations (3 on left isoclines, and 3 on right isoclines), and 5 (or 4) vertices moving over all 24 vertices on isoclines of square fibrations (4 (or 2) on right isoclines, and 1 (or 2) on left isoclines). The up 16-cells already have two vertices moving on octagram isoclines in the minimal configuration, but the down 16-cells have only one. We can add two vertices moving on a chiral pair of isoclines without breaking the charge balance. Adding two such left-right pairs to each down 16-cell leaves the down 16-cell with two up+right and three down−left vertices. It could decay into an up 16-cell (two up+right vertices) by losing the three down−left vertices with a combined unit negative charge, like the electron emitted during [[W:beta decay|beta minus decay]]. This configuration of a single stable proton (or single unstable neutron) has 9 (or 12) vertices moving on isoclines of square fibrations: 6 (or 6) on right isoclines, and 3 (or 6) on left isoclines. The fact that a proton and neutron form a stable nuclide suggests that they can occupy the same 24-cell together, where their moving vertices combine to stabilize the down 16-cell. This is possible because there is enough room in each 16-cell for it to contain the essential moving vertices of both an up and a down quark at the same time: two vertices moving on an up+right isocline, plus one vertex moving on a down−left isocline provided it is not the complement of either of the right rotations (because the left-right pair of the same distinct rotation have exactly opposite isospin and would cancel each other's charge contribution). This [[W:deuterium|deuterium]] configuration of a stable proton-neutron pair has 9 vertices moving on isoclines of square fibrations: 6 on right isoclines, and 3 on left isoclines. It could be created by [[W:beta plus decay|beta plus decay]] when two protons are forced to occupy the same 24-cell, during the first step in the [[W:proton-proton chain|proton-proton chain]] of [[W:Stellar nucleosynthesis|nucleosynthesis]]. == Wiki researchers == [[W:User:Cloudswrest/Regular_polychoric_rings]], A.P. Goucher (https://cp4space.wordpress.com/2012/09/27/good-fibrations/) [[W:User:Tomruen/Uniform_honeycombs]] [[W:User:PAR]] physicist (stat. mech.) ([[wikipedia:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] in [[wikipedia:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]) [[W:User:DGG]], David Goodman (wikipedia admin, librarian, expert on scientific publishing and open access to published research, first amendment absolutist) [[User:Fedosin|S.Fedosin]] e.g. [[SPФ symmetry]], [[Scale dimension]], '''Model of Gravitational Interaction'''<ref>{{Cite web|url=http://sergf.ru/mgen.htm|title=Model of Gravitational Interaction in the Concept of Gravitons|website=sergf.ru|access-date=2019-05-11}}</ref>, [[Hydrogen system]], [[Metric theory of relativity]] - perhaps begin here<ref>http://vixra.org/pdf/1209.0110v1.pdf</ref> === Fibrations === Currently titled '''Visualization''', perhaps this section could be renamed and multiple references given for the [[W:Hopf fibration]]{{Efn|On a 2-sphere (globe), if you go off in any direction and keep going straight you eventually arrive back at your starting point. Same with the 3-sphere, except you are no longer restricted to a plane, you can go off in any 3-dimensional direction. For the 2-D case all great circles intersect. You can avoid this for the 3-D case. Step away from your initial starting point and go off in a new direction. You want to pick this direction so that you don't intersect the previous geodesic. To this end you have to give your new direction a little "skew" so that your new starting direction is not exactly parallel, and out of the plane, to your old direction. This avoidance of intersection causes the two loops/geodesics to spiral around each other and/or interlock. For the Hopf fibration the farther away you are from the initial starting point, the more "skew" you add. When you are 90 degrees away, you add 90 degrees of skew. This is the most extreme case and you have two interlocking rings passing through the midpoint of the other ring. With this construction you can parameterize the whole 3-sphere, with no two rings ever touching each other. ---- [[W:User:Cloudswrest]] in [[W:Talk:Hopf fibration#Untitled|Talk:Hopf fibration]]|name=|group=}} re: the 24-cell. But beware [[W:User:Cloudswrest/Regular polychoric rings|similar material]] by [[W:User:Cloudswrest]] and [[W:User:Tomruen]] was deleted from the [[W:Hopf fibration]] article once for lack of references. {{Efn|This section was deleted for lacking references. I had been adding some graphics. I copied the section to User:Tomruen/Regular polychoric rings. Tom Ruen (talk) 04:01, 15 November 2014 (UTC) I started the "Discrete Examples" section because of the 120-cell. I consider it a perfect example of the Hopf fibration in a different context. A "physical" example. People might not "get it" when given equations, or theory, or even a continuum picture, but seeing the 120-cell example might provide an "aha moment"! You can SEE it in the Todesco Youtube video of the 120-cell. The other face-to-face cases quickly became obvious. Then somebody mentioned, without any references, the BC helices in the 600-cell, and all the tet based polytope fibrations fell into place also. For the most part math articles have been a pretty safe subject to edit as it's objective, self documenting, except for very esoteric stuff, and people who don't know anything about it are usually uninterested, unlike articles on subjects like say, Martin Luther King, or Hitler, or date rape, where all the social justice warriors and polemicists come out to play. When Eppstein first complained about references over a year ago I did a web search. There are bits and pieces on various web sites and blogs, including some on John Baez's blog, but I could not find any coherent full coverage of the subject, which in any case is pretty obvious to interested parties. But I do know that making snide and sarcastic remarks on the rather competent and prolific work of a Wikipedia math illustrator is over the top. Cloudswrest (talk) 02:00, 17 November 2014 (UTC)}} [https://cp4space.wordpress.com/2012/09/27/good-fibrations/ Goucher] [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] [http://members.home.nl/fg.marcelis/mathemathics.htm Marcelis] The first illustration below shows a torus surface on which 4 equidistant circles lie, each having 4 equidistant points that are 4 of the 16 vertices of a hypercube in stereographic projection. On the vertical line and the circle lie 4 equidistant points each, completing the 16 vertices of the hypercube to the 24 vertices of a 24-cell. === Curiosities === https://physics.info/motion/ https://hexnet.org/ [[w:Double bubble conjecture|Double bubble conjecture]] http://vixra.org/pdf/1812.0482v1.pdf perhaps follows Steinbach's polygonal chord relationships === Review === [https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/ Baez] [http://eusebeia.dyndns.org/4d/bi24dim600cell Who is this?] [http://www.cs.utah.edu/~gk/peek/600slice/index.html Peek software] Who is this? http://eusebeia.dyndns.org If identified perhaps could put under External links on some 4-polytope pages. Especially [http://eusebeia.dyndns.org/4d/vis/vis 4D Visualization] [http://members.home.nl/fg.marcelis/ Marcelis] also [https://fgmarcelis.wordpress.com/ Macelis's other website] === Communities === [http://hi.gher.space/forum/ Higher Space Forum] == 4-space generally == === Dimensional analogy opinion === {{Efn|A [[W:Four-dimensional space#Dimensional analogy|dimensional analogy]] is not a metaphor that we are free to adopt or replace, like the conventional names of the 4-polytopes. The 600-cell is the unique 4-dimensional analogue of the icosahedron in a precise mathematical sense. The symmetry group of the 600-cell is only sometimes called the [[W:Binary icosahedral group|binary ''icosahedral'' group]] (by metaphorical analogy), but the dimensional relationship between the 600-cell and the icosahedron which the operations of the group capture is a mathematical fact (a dimensional analogy). It is not a mistake to call the 600-cell the hexacosichoron or the 4-120-polytope or any other reasonably analogous name we may invent, but it would be a mathematical error to misidentify the 600-cell as the analogue of some other polyhedron than the icosahedron.{{Efn|It is important to distinguish ''dimensional'' analogy from ordinary ''metaphorical'' analogy. ''Dimensional analogy''{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process that can function as a guide to proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction.|name=dimensional analogy}}|name=4-dimensional analogue of the icosahedron|group=}} {{Efn|It is a mistake to confuse the finite mathematics of ''dimensional analogy'' with the infinite art of ''metaphorical analogy''. Dimensional analogy{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process, like a proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction. [https://www.npr.org/books/titles/138359394/what-we-believe-but-cannot-prove-todays-leading-thinkers-on-science-in-the-age-o I believe, but I cannot prove], that there is but one ''correct'' dimensional analogy in every instance; moreover, there is ''always'' that one correct dimensional analogy in every instance (though it may well not have been discovered yet).|name=dimensional analogy}} === Words === [http://os2fan2.com/gloss/index.html The polygloss] Wendy Krieger's glossary of higher-dimensional terms. === Math === The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] {1, ''ξ''₁, ''η'', ''ξ''₂} [Coxeter 1973 p. 216]. Formulas for the conversion of Cartesian coordinates to Hopf coordinates: https://marc-b-reynolds.github.io/quaternions/2017/05/12/HopfCoordConvert.html In 3D every displacement can be reduced to a single-rotation combined with a translation (a screw-displacement). In 4D every displacement can be reduced to either a double rotation or a a single rotation combined with a translation (a 3D screw-displacement).[Coxeter 1973 p 218][[File:Pythagorean theorem - Ani.gif|thumb|caption=(3,4,5) is the smallest [[W:Pythagorean triple|Pythagorean triple]] (a [[W:Special right triangle#Side based|special right triangle]])]] The [[W:Icosian|Icosians]] and [[W:Quaternions]] generally, the "3<small>{{Sfrac|1|2}}</small>-dimensional coordinates of projective 4-space [[https://books.google.com/books?id=5-UlBQAAQBAJ&pg=PA207&lpg=PA207&dq=icosian+ring+golden+field&source=bl&ots=_bR1ndRqQ0&sig=ACfU3U1ePHtwHLEfyG5IC0bQr8Ur736hPw&hl=en&sa=X&ved=2ahUKEwjUiKSPzoriAhWEjp4KHYclD6EQ6AEwCXoECAgQAQ#v=onepage&q=icosian%20ring%20golden%20field&f=false|from google books]] Application of quaternions and projective space generally in the vertex figure space (the curved boundary space of a 4-polytope from the inside). [http://eusebeia.dyndns.org/4d/genrot.pdf Formula for Vector Rotation in Arbitrary Planes] [https://www.facebook.com/Formule.byBNF/posts/httpeusebeiadyndnsorg4dindex/10151772568869171/ Viviani's Theorem] In an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the altitude of the triangle [https://fgmarcelis.wordpress.com/ Macelis's other website] on the geometrical mathematics of physics. === Truncation === What does truncation look like from vertex space (the curved 3-manifold)? Vertices are removed like voids carved out of the interior of the 3-space, or rather, the location of the 3-manifold moves in the 4th direction. How can we imagine observing this (as a continuous process) from the inside of the 3-space? == 120-cell == "Who ordered that?"{{Efn|As the Nobel laureate physicist [[W:Isidor Isaac Rabi|I. I. Rabi]] famously quipped about the unanticipated muon, "Who ordered that?".}} [http://eusebeia.dyndns.org/4d/news2014q1 Omnitruncated 120-cell] - the largest uniform convex 4-polytope. [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] and [https://homepages.warwick.ac.uk/~masgar/Maths/quintessence.pdf Schleimer] A first way to understand the combinatorics of the 120–cell is to look at the layers of dodecahedra at fixed distances from the central dodecahedron. A second way to understand the 120–cell is via a combinatorial version of the [[W:Hopf fibration]]. === Falsified theory === To generate the 120-cell from the 600-cell, it is sufficient to rotate the 600 tetrahedra once, through five positions (either the left-handed or the right-handed chiral rotation). Both chiral rotations are not required, because each 5-click rotation by itself generates all the cells of five ''disjoint'' 600-cells, which together comprise all the vertices of the 120-cell and all ten 600-cells. In other words, the two ways to pick five disjoint 600-cells (out of the ten ''distinct'' 600-cells) correspond to the two sets of opposing tetrahedra in each dodecahedron. FALSIFIED{{Sfn|van Ittersum|2020|loc=§4.3.4 Quaternions with real part 1/2 in each 24-cell in the 600-cell 2I|pp=85-86}} === Dodecahedron coordinates === The red vertices lie at (±φ, ±{{sfrac|1|φ}}, 0) and form a rectangle on the ''xy''-plane. The green vertices lie at (0, ±φ, ±{{sfrac|1|φ}}) and form a rectangle on the ''yz''-plane. The blue vertices lie at (±{{sfrac|1|φ}}, 0, ±φ) and form a rectangle on the ''xz''-plane. (The red, green and blue coordinate triples are circular permutations of each other.) === 30-tetrahedron rings are duals of Petrie polygons === [[W:Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons|Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons]] == 600-cell == === Rotations === In the 600-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at most 10 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[digon]], and the completely orthogonal fixed plane intersects 0 vertices, 2 vertices (a digon), 4 vertices (a square), or 6 vertices (a hexagon), respectively. === Misc === [[User:Dc.samizdat/600-cell]] The 120 vertices can be seen as the vertices of four sets of 6 orthogonal equatorial pentagons which intersect only at their common center.{{Efn|The edges of the 600-cell form geodesic (great circle) decagons. One can pick out six orthogonal decagons, lying (for example) in the six orthogonal planes of the 4-axis coordinate system. Being completely orthogonal, these decagons share no vertices (they 'miss' each other and intersect at only one point, their common center). Thus they comprise 60 distinct vertices: half the vertices of the 600-cell. By symmetry, the other 60 vertices must occur in an exactly similar (congruent) configuration as another set of six orthogonal decagons (rotated isoclinically with respect to the first set).|name=orthogonal decagons}}{{Efn|Each decagon in the orthogonal set of 6 must share two vertices (a common diameter) with each decagon to which it is not orthogonal (namely, the 66 decagons not in the set). So each set of 6 (orthogonal) decagons populates vertices in 66 (other) decagons. There are 12 sets of 6 orthogonal decagons.}} {{Efn|For any fixed value of 𝜂, we have a 𝜉_''i'' decagon and a 𝜉_''j'' decagon with disjoint vertex sets, because they are completely orthogonal. Conversely, the 6 decagons which intersect at each vertex cannot be mutually orthogonal, and each must have a different value of 𝜂.}} The 600 tetrahedral cells can be seen as the result of a 5-fold subdivision of 24 octahedral cells yielding 120 tetrahedra, in a compound made of 5 such subdivided 24-cells (rotated with respect to each other in angular units of {{sfrac|𝜋|5}}). The 600-cell's edge length is ~0.618 times its radius (the 24-cell's edge length). This is 𝚽, the smaller of the two golden sections of √5. Its reciprocal, the larger golden section, is φ = 1.618. A {{radic|5}} chord will not fit in a polytope of unit radius ({{radic|4}} diameter), but both of its golden sections will fit, and both occur as vertex chords of the unit-radius 600-cell: the smaller 𝚽 as its edge length, and the larger φ as the chord joining vertices that are 3 edge lengths apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 4 sets of) [[wikipedia:24-cell#Hexagons|4 orthogonal hexagons]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 such disjoint sets of 4 orthogonal hexagons will account for all 120 vertices. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 3 sets of) [[wikipedia:24-cell#Squares|6 orthogonal squares]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 disjoint sets of 6 such orthogonal squares will account for all 120 vertices. Notice the pentagon inscribed in the decagon. Its {{radic|1.𝚫}} edge chord falls between the {{radic|1}} hexagon and the {{radic|2}} square. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges, evidently dodecahedra), which falls between the 24-cells' envelopes of octahedra (made of {{radic|1}} hexagon edges) and the 8-cells' envelopes of cubes (made of {{radic|2}} square edges). Consider also the {{radic|2.𝚽}} = φ and {{radic|3.𝚽}} chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell.{{Efn|1=The {{radic|2.𝚽}} = <big>φ</big> and {{radic|3.𝚽}} chords produce irregular interior faces and cells, since they make isosceles great circle triangles out of two chords of their own size and one of another size.|name=isosceles chords|group=}} The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell, it also inserts new features deep in the interstices of the [[wikipedia:24-cell#Constructions|24-cell's interior]] structure (which it inherits in full, compounds five-fold, and then elaborates on). evidently the 600-cell has dodecahedra in it golden triangles{{Efn|A [[W:Golden triangle|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ration|golden ratio]] to the distinct side ''b'': : {{sfrac|a|b}} <nowiki>=</nowiki> ϕ <nowiki>=</nowiki> {{sfrac|1 + {{radic|5}}|2}} <nowiki>≈</nowiki> 1.618 It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center.<br> The vertex angle is: : <nowiki>𝛉 = arccos(</nowiki>{{sfrac|ϕ|2}}<nowiki>) = </nowiki>{{sfrac|𝜋|5}}<nowiki> = 36°</nowiki> so the base angles are each {{Sfrac|2𝜋|5}} <nowiki>=</nowiki> 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=golden triangle}} ==== Rotations ==== {{Efn|This is another aspect of the same pentagonal symmetry which permits the partitioning of the 600-cell into [[#Icosahedra|icosahedral clusters]] of 20 cells and clusters of 5 cells.}} Each isoclinic rotation occurs in two chiral forms: there is a Clifford parallel 24-cell to the ''left'' of each 24-cell, and another Clifford parallel 24-cell to its ''right''. The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. ==== Central planes ==== All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart in the 600-cell), hexagon planes ({{sfrac|𝜋|3}} apart in each of 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart in each of 75 inscribed 16-cells). In a 4-polytope, two different central planes may intersect at a common diameter, as they would in 3-space, or they may intersect at a single point only, at the center of the 4-polytope. In the latter case, their great circles are [[W:Clifford parallel|Clifford parallel]].{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space.}} Completely orthogonal{{Efn|name=completely orthogonal planes}} great circles are an example of Clifford parallels, but we can also find non-orthogonal central planes which intersect at only a single point. {{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Because they share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3-dimensions; in fact they are connected like adjacent links in a chain, each passing through the other (without intersecting at any points), forming a [[W:Hopf link|Hopf link]]. === Cayley’s Factorization of 4D Rotations === Any rotation in ℝ⁴ can be seen as the composition of two rotations in a pair of orthogonal two-dimensional subspaces. When the values of the rotation angles in these two subspaces are equal, the rotation is said to be isoclinic. Cayley realized that any rotation in ℝ⁴ can be factored into the commutative composition of two isoclinic rotations.<ref>{{Cite journal|last=Perez-Gracia|first=Alba|last2=Thomas|first2=Federico|date=2016-05-14|title=On Cayley’s Factorization of 4D Rotations and Applications|url=http://dx.doi.org/10.1007/s00006-016-0683-9|journal=Advances in Applied Clifford Algebras|volume=27|issue=1|pages=523–538|doi=10.1007/s00006-016-0683-9|issn=0188-7009}}</ref> The elements of the Lie group of rotations in four-dimensional space, SO(4), can be either simple or double rotations. Simple rotations have a fixed plane (a plane in which all the points are fixed under the rotation), while double rotations have a single fixed point only, the center of rotation. In addition, double rotations present at least a pair of invariant planes that are orthogonal. The double rotation has two angles of rotation, α1 and α2, one for each invariant plane, through which points in the planes rotate. All points not in these planes rotate through angles between α1 and α2. Isoclinic rotations are a particular case of double rotations in which there are infinitely many invariant orthogonal planes, with same rotation angles, that is, α1 = ±α2. These rotations can be left-isoclinic, when the rotation in both planes is the same (α1 = α2), or right-isoclinic, when the rotations in both planes have opposite signs (α1 = −α2). Isoclinic rotation matrices have several important properties: # The composition of two right- (left-) isoclinic rotations is a right- (left-) isoclinic rotation. # The composition of a right- and a left-isoclinic rotation is commutative. # Any 4D rotation can be decomposed into the composition of a right and a left-isoclinic rotation. Hence both form maximal and normal subgroups. Their direct product is a double cover of the group SO(4), as four-dimensional rotations can be seen as the composition of rotations of these two subgroups, and there are two expressions for each element of the group. === 30-gon geodesic === The [[w:600-cell#Geodesics|30-gon vertex-less geodesic of the 600-cell]] reminds me of another remarkable observation about the central axis of the B-C helix made many years ago by the dutch software engineer and geometry experimenter Gerald de Jong,{{Efn|I don't recall de Jong ever writing about 4-dimensional polytopes, but he has a large body of work experimenting with physical and virtual models of geodesics and especially tensegrity structures.}} on a long-extinct email list called ''Synergetics'' that mostly featured discussions of Buckminster Fuller{{Efn|Buckminster Fuller never quite got his mind around 4-polytopes, despite knowing Coxeter, but much of what he observed about the 3-polytopes is directly relevant to the 4-polytopes and original; he had splendid intuition. For example in his obsession with the cuboctahedron (which he called the "vector equilibrium") he was probably the first to sense the real importance of the [[w:24-cell#Radially equilateral honeycomb|radially equilateral]] polytopes. Looking at the 24-cell and tesseract makes me sad for him that he never realized the fourth dimension has a ''regular'' vector equilibrium, one of ''two'' radially equilateral regular polytopes (the other of which is the hypercube!). Just as Fuller's studies and those he inspired (such as [[http://verbchu.blogspot.com/2010/07/ccp-and-hcp-family-of-structures.html%7Cthis]]) are often relevant to the 4-polytopes, the 4-polytopes now inspire new 3-dimensional inventions, such as new forms of Fuller's geodesic domes{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} memes. }} That list didn't extend to 4-polytopes; the geometry discussed there was about 3-dimensional objects, as it also tended to be on [[Magnus Wenninger]]'s ''Polyhedron'' email list. I can't find an archived copy of the email list with Gerald's post but as I recall he studied the B-C helix (Fuller called it the ''tetrahelix'') in 3 dimensions and observed that it had no single central axis, but rather three central axes that passed through each tetrahedron similarly, hitting the volume center of the tetrahedron and hitting two faces near but not at their center, like three holes punched in the face in a small equilateral triangle surrounding the face center. He called the tetrahedra pierced by the three central axes "tetrahedral salt cellars", a wonderfully evocative image and why I have remembered it (correctly, I hope). It is interesting to see that when the helix is bent in the fourth dimension into a ring, in addition to its period being rationalized and its helical edge-paths being straightened into geodesics, its three center axes also merge into one, which passes through a single point at the center of each face. === Golden triangles === The [[W:Golden triangle (mathematics)|golden triangle]] is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions. [[File:Golden_Triangle.svg|right|thumb|A golden triangle. The ratio a:b is equivalent to the golden ratio φ.]] [[w:Golden triangle|Golden triangles]] are found in the nets of several stellations of dodecahedrons and icosahedrons. Since the angles of a triangle sum to 180°, base angles are therefore 72° each.^[1] The golden triangle can also be found in a regular decagon, or an equiangular and equilateral ten-sided polygon, by connecting any two adjacent vertices to the center. This will form a golden triangle. This is because: 180(10-2)/10=144 degrees is the interior angle and bisecting it through the vertex to the center, 144/2=72.^[1] [[File:Kepler_triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the [[w:Golden_ratio|golden ratio]].]] A [[w:Kepler_triangle|Kepler triangle]] is a right triangle with edge lengths in a geometric progression in which the common ratio is √φ, where φ is the golden ratio,^[a] and can be written: , or approximately '''1 : 1.272 : 1.618'''.^[1] The squares of the edges of this triangle are in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between its short side and hypotenuse equal to the golden ratio.^[2] Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed:<blockquote>Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.^[3]</blockquote>Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza,^[4][5] making it a golden pyramid. === Golden chords === The ''golden chords'' demonstrate that <big>ϕ</big> is a circle ratio like <big>𝜋</big>, in fact:<br> : {{sfrac|𝜋|5}} = arccos ({{sfrac|ϕ|2}}) which is one decagon edge. Inversely:<br> : <big>ϕ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})<br> which can be seen from the arc length of the {{radic|2.𝚽}} = <big>ϕ</big> golden chord which is {{sfrac|3𝜋|5}}, but it was apparently discovered first without recourse to geometry.<ref>{{Cite web|title=Pi, Phi and Fibonacci|date=May 15, 2012|author=Gary Meisner |url=https://www.goldennumber.net/pi-phi-fibonacci/|postscript=: Robert Everest discovered that you can express <big>ϕ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series: <big>ϕ</big> = 1 – 2 cos (3𝜋/5)}}</ref> Phi is a circle ratio, like Pi Pi = 5 arccos (.5 Phi) Note: The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians. Note: Above formulas expressed in radians, not degrees Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows: 5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series: Phi = 1 – 2 cos ( 3 Pi / 5) Golden ratio of chords: Peter Steinbach * Golden Fields - DPF formula https://www.jstor.org or https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1997.11996494 * Sections Beyond Golden https://archive.bridgesmathart.org/2000/bridges2000-35.pdf Sacred cut (of octagon) https://archive.bridgesmathart.org/2011/bridges2011-559.pdf [[W:George Phillips Odom Jr.]] discovery of golden section in the mid-edge-bisectors of the tetrahedron (applies to the construction of the 600-cell from the 24-cell via truncation of its central tetrahedra) - also look at the other circle/triangle and sphere/tetrahedra relationships he discovered - footnote his relationship to Coxeter and Conway [[w:Intersecting chords theorem|Intersecting chords theorem]] === Nonconvex regular decagon === [[File:Golden_tiling_with_rotational_symmetry.svg|left|thumb|This '''[[Tessellation|tiling]]''' by '''[[Golden triangle (mathematics)|golden]]''' triangles, a regular '''[[pentagon]]''', contains a '''[[wikipedia:Stellation|stellation]]''' of '''[[Regular polygon|regular]] decagon''', the '''[[Schläfli symbol|Schäfli&nbsp;symbol]]''' of which is {10/3}.]] The length [[ratio]] of two inequal edges of a golden triangle is the [[golden ratio]], denoted <math>\text{by }\Phi \text{,}</math> or its [[Multiplicative inverse|multiplicative&nbsp;inverse]]: :<math> \Phi - 1 = \frac{1}{\Phi} = 2\,\cos 72\,^\circ = \frac{1}{\,2\,\cos 36\,^\circ} = \frac{\,\sqrt{5} - 1\,}{2} \text{.}</math> So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this [[Star polygon|star&nbsp;polygon]].{{Clear}} == 24-cell == Visualize the three 16-cells inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} [[File:24-cell graph D4.svg|thumb]] Whoever this is has some beautiful illustrations of closely related polytopes e.g.<ref>http://eusebeia.dyndns.org/4d/rect24cell</ref> [http://members.home.nl/fg.marcelis/24-cell.htm Marcelis] including [http://members.home.nl/fg.marcelis/24celiso.htm#gem the gem of the modular universe] === Cuboctahedron === In the 24-cell, there are 16 great hexagons, and each has one pair of vertices (one diameter) which lies on a Cartesian coordinate system axis (in the vertex-up frame of reference). The 24-cell can be seen as the 24 vertices of four orthogonal great hexagons, each aligned with one of the four orthogonal coordinate system axes, and each contributing 6 disjoint vertices. In four dimensions, the four ''orthogonal'' planes do not intersect except at their common center. Each great hexagon does not share any vertices with the 3 other hexagons to which it is orthogonal,{{Efn|In four dimensions up to 6 planes through a common point may be mutually orthogonal. The 18 great squares of the 24-cell comprise three sets of 6 orthogonal planes. The 16 great hexagons, however, comprise four sets of just 4 orthogonal planes. In four dimensions there is both a symmetrical arrangement of 6 orthogonal planes, and a symmetrical arrangement of 4 orthogonal planes. We can pick out 6 orthogonal squares in the 16-cell, 8-cell, or 24-cell, but the symmetry of 4 orthogonal hexagons emerges only in the 24-cell.}} but it shares two vertices with each of the 12 other great hexagons to which it is ''not'' orthogonal. Four hexagonal geodesics pass though each vertex. In the cuboctahedron, there are four great hexagons, but they are not orthogonal. Each intersects with each of the others in two vertices (one diameter), and two hexagons pass through each vertex. (In three dimensions, two planes through a common point intersect in a line, whether they are orthogonal or not.) At most one of the four hexagons can have a pair of vertices (a diameter) which lies on a coordinate system axis. In such a frame of reference, the other two axes pass through the centers of a pair of opposing triangular faces, and through the centers of a pair of opposing edges, respectively. === Geometry === ==== Triangles ==== ...to be added:<br> If the dual of the [[W:24-cell#Squares|24-cell of edge length {{radic|2}}]] is taken by reciprocating it about its ''circumscribed'' sphere, another 24-cell is found which has edge length and circumradius {{radic|3}}, and its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows: :<math>(0, \pm 1, \pm 1, \pm 1) \in \mathbb{R}^4</math> The 4 orthogonal planes in which the 8 triangles lie are ''not'' orthogonal planes of this coordinate system. The triangles' {{radic|3}} edges are the ''diagonals'' of cubical ''cells'' of this coordinate lattice.{{Efn|For example: {{green|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> are the two opposing central triangles on the ''y'' axis (in this coordinate system of edge length {{radic|3}}).|name=|group=}} The 24 vertices are also the vertices of 96 ''other'' triangles of edge length {{radic|3}} that occur in 48 parallel pairs, in planes one edge length apart. Each plane sections the 24-cell through three vertices but does not pass through the center.{{Efn|...add coordinates example to existing note...|name=|group=}} ..see [[w:5-cell#Construction|5-cell#Construction]]: Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}}: :<math>\left( 1,1,1, \frac{-1}\sqrt{5} \right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,-1,1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0,0,0,\sqrt{5}-\frac 1\sqrt{5} \right)</math> ... obs: <br> Add coordinate examples to this footnote{{Efn|Each of these 96 triangular planes sections the 24-cell <small>{{sfrac|1|2}}</small> edge-length below a vertex, and <small>{{sfrac|1|2}}</small> edge-length above the center, measured from the center of the triangle, which is on a 24-cell diameter joining two opposite vertices. However, these paired parallel triangular planes are not orthogonal to the diameter line; they are inclined with respect to it. Each plane contains only one triangle (unlike the central hexagonal planes with their two opposing triangles), but they occur in co-centric sets of four, inclined different ways about the diameter line. The 96 triangles are inclined both with respect to the coordinate system's 6 orthogonal planes (the 6 perpendicular squares) and with respect to the hexagons. Each triangle contains one vertex from a square, and two from different hexagons. Thus their Cartesian coordinates take many different forms, but as examples: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> is one of the four face triangles with one vertex at the positive vertex on the ''y'' axis; and below is the opposite face of one of the two tetrahedra it is a face of, inclined about the negative part of the ''y'' axis: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> |name=non-central triangles}} === Rotations === The vertices of a convex 4-polytope lie on the [[W:3-sphere|3-sphere]], so alternatively they can be described using 4-dimensional spherical coordinates such as the [[w:N-sphere#Hopf_coordinates|Hopf coordinates]] {''r'', ''ξ''₁, ''η'', ''ξ''₂}, which reveal more structure. For the 24-cell of edge length and radius 1, the Hopf coordinates of its vertices can be obtained by permuting the three angle coordinates as follows: {1, ±{{sfrac|𝜋|3}}, ±{{sfrac|𝜋|2}}, ±{{sfrac|2𝜋|3}}}[[File:30-60-90.svg|thumb]] in 3D section of a 24-cell (of edge length 2) we can construct a tetrahedron with 2{{radic|3}} legs descending from a vertex (0,0,2) with its base plane triangle intersecting the 2-sphere at (x,y,-1), but only if we put the base vertices at distances apart less than 2{{radic|3}}, as a 2{{radic|2}} 3 3 isosceles triangle e.g. with these vertices: (-{{radic|2}},-1,-1) <--- 2{{radic|2}} --> ((+{{radic|2}},-1,-1)) <-- 3 --> (0, (+{{radic|3}}, -1) <-- 3 --> Apparently the base vertices of the tetrahedron are displaced out of this hyperplane in the 4th dimension so the base edges are foreshortened. ... and eight meet at the volume center of each tesseract cube{{Efn|The geometry of the tesseract cube volume centers is exactly the same as the vertices: a cubical vertex figure, in which four long diameters cross at the center. They are 24 interior vertices, arrayed as a 1/2 size 24-cell around the central interior vertex, at the midpoints of the 24 unit-length radii (which pass through opposite face centers of the vertex figure).}} {{Clear}} ==Kepler problem== [[File:Kepler-solar-system-2.png|thumb|Detailed view of the inner sphere of Kepler's Mysterium Cosmographicum.]]We are apt to be smug about the quaint mythological phantasies of our great forebears, as when we learn that Issac Newton worked for more than 30 years as an alchemist trying to turn base metals into gold, before he was appointed Chancellor of the Exchequer to preside over England's mint of sterling instead. [[W:Mysterium Cosmographicum|Kepler's astrolabe]] of Plato's holy solids looks that way to us, like a religious miracle the great astronomer hoped for, before he discovered the real symmetry in his three great conservation laws of motion. It should humble us, then, to find out that he was on to something deeper all along, and make us wonder all the more at his genius, that it was the SO(4) rotational symmetry he glimpsed, which generates those conservation laws by Noether's theorem, and also generates the 4-polytopes that Schläfli would discover on the 3-sphere in 4-space two and a half centuries after Kepler, who somehow imagined them from below, projected on the 2-sphere in 3-space. ==Laplace–Runge–Lenz vector== the following rescued from [[W:Laplace–Runge–Lenz vector]] version of 23:32, 27 November 2006 from which it was removed by WillowW (talk | contribs) at 23:46, 28 November 2006 (the Moebius transform was fun, but needs to go now) ===Intuitive picture of the rotations in four dimensions=== [[Image:Kepler_hodograph_family_transformed.png|thumb|right|280px|Figure 4: [[W:Circle inversion|Inversion]] in the dotted black circle of Figure 3 transforms the family of circular hodographs of a given energy ''E'' into a family of straight lines intersecting at the same point. Thus, the orbits of the same energy but different angular momentum can be transformed into one another by a simple rotation.]] The simplest way to visualize the particular symmetry of the Kepler problem is through its [[W:hodograph|hodograph]]s, the perfectly circular traces of the momentum vector (Figures 2 and 3). For a given total [[W:energy|energy]] ''E'', the hodographs are circles centered on the ''p_y''-axis, all of which intersect the ''p_x''-axis at the same two points, ''p_x=±p₀'' (Figure 3). To eliminate the normal rotational symmetry, the coordinate system has been fixed so that the orbit lies in the ''x''-''y'' plane, with the major semiaxis aligned with the ''x''-axis. [[W:Circle inversion|Inversion]] centered on one of the foci transforms the hodographs into straight lines emanating from the inversion center (Figure 4). These straight lines can be converted into one another by a simple two-dimensional rotation about the inversion center. Thus, all orbits of the same energy can be continuously transformed into one another by a rotation that is independent of the normal three-dimensional rotations of the system; this represents the "higher" symmetry of the Kepler problem. == Formatting idioms == :{{sfrac|1|2}}(±φ𝞍ϕ𝜙𝝓𝚽𝛷𝜱, ±1, ±{{sfrac|1|φ}}, 0). : :360^{\circ} :denoted <math>\tbinom{24}{4}</math>   == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book)}} * {{Cite book | last1=Coxeter | first1=H.S.M. | author1-link=W:Harold Scott MacDonald Coxeter | last2=du Val | first2=Patrick | author2-link=W:Patrick du Val | last3=Flather | first3=H.T. | last4=Petrie | author4-link=W:John Flinders Petrie | first4=J.F. | year=1938 | title=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]] | publisher=University of Toronto Studies (Mathematical Series) | volume=6 }} * {{Cite book|title=The Coxeter Legacy|chapter=Coxeter Theory: The Cognitive Aspects|last=Borovik|first=Alexandre|year=2006|publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|pp=17-43|ISBN=978-0821837221|url=https://www.academia.edu/26091464/Coxeter_Theory_The_Cognitive_Aspects}} * {{Cite journal | last=Miyazaki | first=Koji | year=1990 | title=Primary Hypergeodesic Polytopes | journal=International Journal of Space Structures | volume=5 | issue=3–4 | pages=309–323 | doi=10.1177/026635119000500312 | s2cid=113846838 }} * {{Cite book|url=https://link.springer.com/chapter/10.1007/978-981-10-7617-6_6|title=Nanoinformatics|last=Nishio|first=Kengo|last2=Miyazaki|first2=Takehide|publisher=Springer|year=2018|isbn=|editor-last=Tanaka|location=Singapore|pages=97-130|chapter=Polyhedron and Polychoron Codes for Describing Atomic Arrangements}} * {{Cite journal|last=Waegell|first=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|url=https://arxiv.org/abs/0911.2289v2|language=en|doi=10.1088/1751-8113/43/10/105304}} * {{Cite journal|last=Sadoc|first=Jean-Francois|date=2001|title=Helices and helix packings derived from the {3,3,5} polytope|journal=[[W:European Physical Journal E|European Physical Journal E]]|volume=5|pages=575–582|doi=10.1007/s101890170040|doi-access=free|s2cid=121229939|url=https://www.researchgate.net/publication/260046074}} {{Refend}} <references group="lower-alpha" /> mfqhgxlqq6yfiragouypwbbw0ol4pib 2691728 2691726 2024-12-13T01:43:18Z Dc.samizdat 2856930 2691728 wikitext text/x-wiki {{User sandbox}} == Polyscheme learning project == The Polyscheme learning project is intended to be series of articles on the regular polytopes and Euclidean spaces of 4 or more dimensions, that expand the corresponding series of Wikipedia encyclopedia articles to book length with textbook-like treatment of the material, additional learning resources and explanatory notes. == David Christie on Tolkien, to Ira Sandperl (emeritus) on Tolstoy == No doubt Tolkien's novels are not as profound as Tolstoy's with respect to our human condition, but Tolkien's address the more general condition of all peoples. He was the first novelist to write about the affairs of many peoples rather than just the comparatively parochial affairs of men. There is an expansive diversity to his interests, which shows his concern to transcend not just the patriarchy, but the humanitarchy. It disproves his critics' claims that his focus is not wide enough to include women, or people of color, or domains of experience (such as sex) which the critics feel are not sufficiently represented in his stories. == Regular convex 4-polytopes == {{W:Template:Regular convex 4-polytopes|wiki=W:}} {{Regular convex 4-polytopes|wiki=W:}} {| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes |- !align=right|[[W:Symmetry group|Symmetry group]] |align=center|[[W:Coxeter_group|A₄]] |align=center colspan=2|[[W:Hyperoctahedral_group|B₄]] |align=center|[[W:F4_(mathematics)|F₄]] |align=center colspan=2|[[W:H4_polytope|H₄]] |- !valign=top align=right|Name |valign=top align=center|[[W:5-cell|5-cell]]<BR> <BR> Hyper-<BR> [[W:Tetrahedron|tetrahedron]] |valign=top align=center|[[W:16-cell|16-cell]]<BR> <BR> Hyper-<BR> [[W:Octahedron|octahedron]] |valign=top align=center|[[W:8-cell|8-cell]]<BR> <BR> Hyper-<BR> [[W:Cube|cube]] |valign=top align=center|[[W:24-cell|24-cell]] |valign=top align=center|[[W:600-cell|600-cell]]<BR> <BR> Hyper-<BR> [[W:Icosahedron|icosahedron]] |valign=top align=center|[[W:120-cell|120-cell]]<BR> <BR> Hyper-<BR> [[W:Dodecahedron|dodecahedron]] |- !align=right|[[W:Schläfli symbol|Schläfli symbol]] |align=center|{3, 3, 3} |align=center|{3, 3, 4} |align=center|{4, 3, 3} |align=center|{3, 4, 3} |align=center|{3, 3, 5} |align=center|{5, 3, 3} |- !align=right|[[W:Coxeter-Dynkin diagram|Coxeter diagram]] |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} |align=center|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} |- !valign=top align=right|Graph |align=center|[[Image:4-simplex t0.svg|120px]] |align=center|[[Image:4-cube t3.svg|121px]] |align=center|[[Image:4-cube t0.svg|120px]] |align=center|[[Image:24-cell t0 F4.svg|120px]] |align=center|[[Image:600-cell graph H4.svg|120px]] |align=center|[[Image:120-cell graph H4.svg|120px]] |- !align=right|Vertices |align=center|5 |align=center|8 |align=center|16 |align=center|24 |align=center|120 |align=center|600 |- !valign=top align=right|Edges |align=center|10 |align=center|24 |align=center|32 |align=center|96 |align=center|720 |align=center|1200 |- !valign=top align=right|Faces |align=center|10<BR>triangles |align=center|32<BR>triangles |align=center|24<BR>squares |align=center|96<BR>triangles |align=center|1200<BR>triangles |align=center|720<BR>pentagons |- !valign=top align=right|Cells |align=center|5<BR>tetrahedra |align=center|16<BR>tetrahedra |align=center|8<BR>cubes |align=center|24<BR>octahedra |align=center|600<BR>tetrahedra |align=center|120<BR>dodecahedra |- !valign=top align=right|[[W:Cartesian coordinates|Cartesian]]{{Efn|The coordinates (w, x, y, z) of the unit-radius origin-centered 4-polytope are given, in some cases as {permutations} or [even permutations] of the coordinate values.}}<BR>coordinates |align=center|{{font|size=85%|( 1, 0, 0, 0)<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}})<BR>(−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})}} |align=center|{{font|size=85%|(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})}} |align=center|{{font|size=85%|({±1, 0, 0, 0})<BR>(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})<BR>([±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0])}} |align=center| |- !valign=top align=right|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf]]{{Efn|name=Hopf coordinates key|[[W:3-sphere#Hopf coordinates|Hopf spherical coordinates]]{{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.{{Sfn|Sadoc|2001|pp=575-576|loc=§2.2 The Hopf fibration of S3}} A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two completely orthogonal invariant planes{{Efn|The point itself (𝜉_''i'', 𝜂, 𝜉_''j'') does not necessarily lie in either of the invariant planes of rotation referenced to locate it (by convention, the ''wz'' and ''xy'' Cartesian planes), and never lies in both of them, since completely orthogonal planes do not intersect at any point except their common center. When 𝜂 {{=}} 0, the point lies in the 𝜉_''i'' "longitudinal" ''wz'' plane; when 𝜂 {{=}} {{sfrac|𝜋|2}} the point lies in the 𝜉_''j'' "equatorial" ''xy'' plane; and when 0 < 𝜂 < {{sfrac|𝜋|2}} the point does not lie in either invariant plane. Thus the 𝜉_''i'' and 𝜉_''j'' coordinates number vertices of two completely orthogonal great circle polygons which do ''not'' intersect (at the point or anywhere else).|name=reference planes of rotation}} which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} The 𝜉_''i'' and 𝜉_''j'' coordinates range over the vertices of completely orthogonal great circle polygons which do not intersect at any vertices. Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<BR> : w {{=}} cos 𝜉_''i'' sin 𝜂<BR> : x {{=}} cos 𝜉_''j'' cos 𝜂<BR> : y {{=}} sin 𝜉_''j'' cos 𝜂<BR> : z {{=}} sin 𝜉_''i'' sin 𝜂<BR> The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).<BR> The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], denoted SO(4), generates those polytopes. A rotation in 4D of a point {{math|{''ξ''_i, ''η'', ''ξ''_j<nowiki>}</nowiki>}} through angles {{math|''ξ''₁}} and {{math|''ξ''₂}} is simply expressed in Hopf coordinates as {{math|{''ξ''_i + ''ξ''₁, ''η'', ''ξ''_j + ''ξ''₂<nowiki>}</nowiki>}}.}} of the vertices are given as three independently permuted coordinates:<BR> :(𝜉_''i'', 𝜂, 𝜉_''j'')_𝑚<BR> where {<''k''} is the {permutation} of the ''k'' non-negative integers less than ''k'', and {≤''k''} is the permutation of the ''k''+1 non-negative integers less than or equal to ''k''. Each coordinate permutes one set of the 4-polytope's great circle polygons, so the permuted coordinate set expresses one set of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-space]] which generates the 4-polytope. With Cartesian coordinates the choice of radius is a parameter determining the reference frame, but Hopf coordinates are radius-independent: all Hopf coordinates convert to unit-radius Cartesian coordinates by the same mapping. {{Efn|name=Hopf coordinates conversion}} Unlike Cartesian coordinates, Hopf coordinates are not necessarily unique to each point; there may be Hopf coordinate synonyms for a vertex. The multiplicity 𝑚 of the coordinate permutation is the ratio of the number of Hopf coordinates to the number of vertices.}}<BR>coordinates |align=center|[[User:Dc.samizdat/sandbox#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle squares of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |align=center|[[User:Dc.samizdat/sandbox#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |align=center|{{font|color=green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}</small>)₁}} |- !valign=top align=right|Long radius{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope.|name=polytopes ordered by size and complexity}} |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |align=center|1 |- !valign=top align=right|Edge length |align=center|<small>{{sfrac|{{radic|5}}|{{radic|2}}}}</small> ≈ 1.581 |align=center|{{radic|2}} ≈ 1.414 |align=center|1 |align=center|1 |align=center|<small>{{sfrac|1|ϕ}}</small> ≈ 0.618 |align=center|<small>{{Sfrac|1|{{radic|2}}ϕ²}}</small> ≈ 0.270 |- !valign=top align=right|Short radius |align=center|{{sfrac|1|4}} |align=center|{{sfrac|1|2}} |align=center|{{sfrac|1|2}} |align=center|<small>{{sfrac|{{radic|2}}|2}}</small> ≈ 0.707 |align=center|<small>1 - ({{sfrac|{{radic|2}}|2{{radic|3}}φ}})²</small> ≈ 0.936 |align=center|<small>1 - ({{sfrac|1|2{{radic|3}}φ}})²</small> ≈ 0.968 |- !valign=top align=right|Area |align=center|<small>10•{{sfrac|{{radic|8}}|3}}</small> ≈ 9.428 |align=center|<small>32•{{sfrac|{{radic|3}}|4}}</small> ≈ 13.856 |align=center|24 |align=center|<small>96•{{sfrac|{{radic|3}}|8}}</small> ≈ 20.785 |align=center|<small>1200•{{sfrac|{{radic|3}}|8φ²}}</small> ≈ 99.238 |align=center|<small>720•{{sfrac|25+10{{radic|5}}|8φ⁴}}</small> ≈ 621.9 |- !valign=top align=right|Volume |align=center|<small>5•{{sfrac|5{{radic|5}}|24}}</small> ≈ 2.329 |align=center|<small>16•{{sfrac|1|3}}</small> ≈ 5.333 |align=center|8 |align=center|<small>24•{{sfrac|{{radic|2}}|3}}</small> ≈ 11.314 |align=center|<small>600•{{sfrac|1|3{{radic|8}}φ³}}</small> ≈ 16.693 |align=center|<small>120•{{sfrac|2 + φ|2{{radic|8}}φ³}}</small> ≈ 18.118 |- !valign=top align=right|4-Content |align=center|<small>{{sfrac|{{radic|5}}|24}}•({{sfrac|{{radic|5}}|2}})⁴</small> ≈ 0.146 |align=center|<small>{{sfrac|2|3}}</small> ≈ 0.667 |align=center|1 |align=center|2 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 3.907 |align=center|<small>{{sfrac|Short∙Vol|4}}</small> ≈ 4.385 |} == Scratch == 400 <math>\sqrt{5} \curlywedge (2-\phi)</math> hexagons In each hemi-icosahedron, 15 edges come from 10 disjoint 5-cells each contributing 4 edges to this hemi-icosahedron, and 3 hemi-icosahedra fitting together around each edge sharing it, as Grünbaum discovered they do. One other hemi-icosahedron fits against each of 10 hemi-icosahedron faces, and two other hemi-icosahedra fit around each of 15 opposite edge, all the same set of 11 hemi-icosahedra. == Coordinate systems on the 3-sphere == The four Euclidean coordinates for {{math|''S''³}} are redundant since they are subject to the condition that {{math|1=''x''₀² + ''x''₁² + ''x''₂² + ''x''₃² = 1}}. As a 3-dimensional manifold one should be able to parameterize {{math|''S''³}} by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as [[latitude]] and [[longitude]]). Due to the nontrivial topology of {{math|''S''³}} it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use ''at least'' two [[coordinate chart]]s. === Hopf coordinates of the regular convex 4-polytopes === As with Cartesian coordinates, there are multiple reference frames which give useful Hopf coordinates.{{Efn|name=Hopf coordinates}} One can choose any of the 4-polytope's great circle polygons for the 𝜉_''i'' coordinate to range over; then the 𝜉_''j'' coordinate will range over the vertices of whatever kind of great circle polygon lies orthogonal to the 𝜉_''i'' great circle plane. Note that the 𝜉_''j'' polygon will sometimes be a digon (a great circle plane intersecting only 2 vertices), as in the case of the the planes orthogonal to the 24-cell's hexagonal planes. The choice of polygons will (almost) determine the only possible range for the 𝜂 coordinate; the only remaining variable is the multiplicity 𝓂 of the coordinates. {| class="wikitable mw-collapsible {{{collapsestate|mw-expanded}}}" style="white-space:nowrap" !colspan=8|[[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Efn|name=Hopf coordinates key}} of the regular convex 4-polytopes |- ! ![[W:5-cell|5-cell]] ![[W:16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[W:24-cell|24-cell]] ![[W:600-cell|600-cell]] ![[120-cell]] |- !Digons |[[#Great circle digons of the 5-cell|{{font color|green|(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀}}]] | | | | |{{font color|green|(<small>{<2}𝜋, {<150}{{sfrac|𝜋|300}}, {<2}𝜋</small>)₁}} |- !Squares | |[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]] | |rowspan=2|[[#Great circle squares of the 16-cell|{{font color|blue|(<small>{<4}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<4}{{sfrac|𝜋|2}}</small>)₄}}]]<BR>[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<15}{{sfrac|𝜋|30}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |{{font color|green|(<small>{<4}{{sfrac|𝜋|2}}, {<75}{{sfrac|𝜋|150}}, {<4}{{sfrac|𝜋|2}}</small>)₂}} |- !Rectangles | | |[[#45 degree axes of the 8-cell|{{font color|blue|(<small>{1 3 5 7}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}</small>)₁}}]] | | !Pentagons | | | | |[[#Great circle pentagons of the 600-cell|{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₅}}]] |{{font color|green|(<small>{0 2 4 6 8}{{sfrac|𝜋|5}}, {<24}{{sfrac|𝜋|48}}, {1 3 5 7 9}{{sfrac|𝜋|5}}</small>)₁}} |- !Hexagons | | | |[[#Great circle hexagons and squares of the 24-cell|{{font color|red|({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<5}{{sfrac|𝜋|10}}, {<4}{{sfrac|𝜋|2}}</small>)₁}} |[[#Great circle squares and hexagons of the 120-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤24}{{sfrac|𝜋|48}}, {<4}{{sfrac|𝜋|2}}</small>)₁}}]] |- ! | | | |[[#Great circle hexagons of the 24-cell|{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, ({<6}{{sfrac|𝜋|3}}</small>)₆}}]] |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}}, {<10}{{sfrac|𝜋|20}}, ({<6}{{sfrac|𝜋|3}}</small>)₃}} |{{font color|green|(<small>{<6}{{sfrac|𝜋|3}},{<50}{{sfrac|𝜋|100}},{<6}{{sfrac|𝜋|3}}</small>)₃}} |- !Decagons | | | | |[[#Great circle decagons and hexagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤1}{{sfrac|𝜋|2}}, {<6}{{sfrac|𝜋|3}}</small>)₁}}]] |{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {<10}{{sfrac|𝜋|20}}, {<6}{{sfrac|𝜋|3}}</small>)₁}} | |- ! | | | | |[[#Great circle decagons of the 600-cell|{{font color|green|(<small>{<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, ({<10}{{sfrac|𝜋|5}}</small>)₅}}]] |[[#Great circle decagons of the 120-cell|{{font color|red|(<small>{<10}{{sfrac|𝜋|5}},{≤5}{{sfrac|𝜋|10}},{<10}{{sfrac|𝜋|5}}</small>)₁}}]] |- !30-gons | | | | | |{{font color|green|(<small>{<30}{{sfrac|𝜋|15}},{≤15}{{sfrac|𝜋|30}},{<30}{{sfrac|𝜋|15}}</small>)₂₄}} |} ==== Great circle digons of the 5-cell==== One set of Cartesian origin-centered [[W:5-cell#Construction|coordinates for the 5-cell]] can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}} and radius {{radic|3.2}}: ( {{sfrac|4|{{radic|5}}}}, 0, 0, 0) (−{{sfrac|1|{{radic|5}}}}, 1, 1, 1) (−{{sfrac|1|{{radic|5}}}}, 1,−1,−1) (−{{sfrac|1|{{radic|5}}}},−1, 1,−1) (−{{sfrac|1|{{radic|5}}}},−1,−1, 1) Rescaled to unit radius and edge length {{sfrac|{{radic|5}}|{{radic|2}}}} these coordinates are: ( 1, 0, 0, 0) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}) (−{{sfrac|1|4}},−{{sfrac|{{radic|5}}|4}},−{{sfrac|{{radic|5}}|4}}, {{sfrac|{{radic|5}}|4}}) {| class="wikitable" !colspan=2|Great circle digons of the 5-cell<BR> Cartesian{{s|2}}({ , , , })<BR> Hopf{{s|2}}(<small>{<2}𝜋, {<30}{{sfrac|𝜋|60}}, {<2}𝜋</small>)₁₂₀ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( 1, 0, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 1{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |(-{{sfrac|1|4}},-{{sfrac|{{radic|5}}|4}}, 0, 0)||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|4}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|5}}|4}} ≈ 0.559}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 3{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |( , , , )||( , , , ) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |( , , , )||( , , , ) !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 4{{sfrac|𝜋|30}}, 𝜉_''j'')||0𝜋||1𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !1𝜋 |{{font|size=75%|( , , , )}}||{{font|size=75%|( , , , )}} !{{font|size=75%|-1}}||{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} | |} ==== Great squares of the 16-cell ==== {| class="wikitable floatright" | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''xy'' plane |- |( 0, 1, 0, 0)||( 0, 0,-1, 0) |- |( 0, 0, 1, 0)||( 0,-1, 0, 0) |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''wz'' plane |- |( 1, 0, 0, 0)||( 0, 0, 0,-1) |- |( 0, 0, 0, 1)||(-1, 0, 0, 0) |} |}The 8 vertices of the 16-cell lie on the 4 coordinate axes and form 6 great squares in the 6 orthogonal central planes. The Cartesian axes lie on the diagonals of the square tables, which resemble the great squares. Rotate the tables 45 degrees clockwise for a vertex up orientation, and another 90 degrees for the standard ''xy'' orientation. By convention rotations are always specified in two completely orthogonal invariant planes xy (whose vertices are numbered counterclockwise by 𝜉_''xy'') and wz (whose vertices are numbered counterclockwise by 𝜉_''wz''). The rotation in the xy plane does not move points in the wz plane, and vice versa. In the 16-cell these two simple rotations rotate disjoint sets of 4 vertices each (because completely orthogonal planes intersect only at the origin and share no vertices). The 𝜂 coordinate of the 4 vertices in the xy plane is 0 and the 𝜂 coordinate of the 4 vertices in the wz plane is 1 ({{sfrac|𝜋|2}}). The w and z coordinates of the vertices in the xy plane are 0 regardless of the rotational position of the wz plane (the 𝜉_''wz'' coordinate), and the x and y coordinates of the vertices in the wz plane are 0 regardless of the rotational position of the xy plane (the 𝜉_''xy'' coordinate). {| class="wikitable" !colspan=2|Great squares of the 16-cell<BR> Cartesian{{s|2}}({0, ±1, 0, 0})<BR> Hopf{{s|2}}(<small>{<3}{{sfrac|𝜋|2}}, {≤1}{{sfrac|𝜋|2}}, {<3}{{sfrac|𝜋|2}}</small>) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''xy'' plane |- !(𝜉_''xy'', 0, 0)||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 0, 1, 0, 0)||( 0, 0,-1, 0) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 1, 0)||( 0,-1, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}0: 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}0: 1}} |} | {| class="wikitable" style="white-space:nowrap;" !colspan=5|''wz'' plane |- !(0, {{sfrac|𝜋|2}}, 𝜉_''wz'')||1{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|2}} |( 1, 0, 0, 0)||( 0, 0, 0,-1) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !2{{sfrac|𝜋|2}} |( 0, 0, 0, 1)||(-1, 0, 0, 0) !{{font|size=75%| 0}}||{{font|size=75%|-1}} |- !{{font|size=75%|sin}}||{{font|size=75%| 1}}||{{font|size=75%|-1}}||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 0}}||{{font|size=75%| 0}}|| ||{{font|size=75%|𝜂{{=}}{{sfrac|𝜋|2}}: 0}} |} |} ====Great rectangles (60 degree planes) of the 8-cell==== None of the 8-cell's 16 vertices lie in the 6 orthogonal central planes. The "north pole" is not a vertex, and 0 does not appear as a value in any (Hopf or Cartesian) coordinate. Each of the 8-cell's eight {{radic|4}} long diameters joining two antipodal vertices lies 45 degrees ({{sfrac|𝜋|4}}) off each of the 4 Cartesian coordinate axes. The Hopf 𝜼 coordinate is {{sfrac|𝜋|4}} for all the vertices, and the 𝜉_''xy'' and 𝜉_''wz'' coordinates are even and odd multiples of {{sfrac|𝜋|4}} respectively. Although the 16 vertices do not lie in the 6 orthogonal central planes, they do lie (by fours) in central planes, but the central polygons they form are rectangles (not squares), and the planes are inclined at 60 degrees ({{sfrac|𝜋|3}}) to each other and to the orthogonal central planes. These 16 great rectangles measure {{radic|1}} by {{radic|3}}, and their {{radic|1}} edges are opposite pairs of 8-cell edges. Their {{radic|3}} edges are interior chords of the 8-cell: long diagonals of the 8 cubic cells.{{Efn|In the 24-cell we find these 16 central rectangles inscribed in the 16 central hexagons. The 8-cell's great rectangles are the same central planes as the 24-cell's great hexagons, but the 8-cell has only 4 of the 6 hexagon vertices. The 16-point (8-cell) is a 24-point (24-cell) with 8 points (square pyramids) lopped off.}} Because there is only one 𝜼 coordinate value, only one table is required, but note that the table is not a great square but a duocylinder (its opposite edges are identified).{{Efn|For fixed {{mvar|η}} Hopf coordinates describe a torus parameterized by {{math|''ξ''_''xy''}} and {{math|''ξ''_''wz''}}, with {{math|''η'' {{=}} {{sfrac|π|4}}}} being the special case of the [[W:Clifford torus|Clifford torus]] in the {{mvar|xy}}- and {{mvar|wz}}-planes. All vertices of the 8-cell lie on the Clifford torus, a "flat" 2-dimensional surface embedded in the 3-sphere. The Clifford torus divides the 3-sphere into two congruent ''solid'' tori. [[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|In a stereographic projection]], the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized.}} Each row, column or diagonal in the table is a great rectangle. {| class="wikitable" ! colspan="2" |Great rectangles (60 degree planes) of the 8-cell Cartesian{{s|2}}(<small>±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}</small>)<BR> Hopf{{s|2}}({0 2 4 6}{{sfrac|𝜋|4}}, {{sfrac|𝜋|4}}, {1 3 5 7}{{sfrac|𝜋|4}}) |- | {| class="wikitable" !(𝜉_''xy'', {{sfrac|𝜋|4}}, 𝜉_''wz'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !0{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} ! {{font|size=75%| {{radic|{{sfrac|1|2}}}}}} |- !2{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !4{{sfrac|𝜋|4}} |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(-<small>{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%| 0}} !{{font|size=75%|−1}} |- !6{{sfrac|𝜋|4}} |(<small> {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) |(<small> {{sfrac|1|2}}, {{sfrac|1|2}},-{{sfrac|1|2}},-{{sfrac|1|2}}</small>) !{{font|size=75%|−1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} ==== Great squares and rectangles of the 24-cell ==== The great square coordinates of the 16-cell (above), combined with the great rectangle coordinates of the 8-cell (above), comprise a set of coordinates for the 24-cell. Because the 16-cell lies vertex-up in this coordinate system, so does the 24-cell. ==== Great squares of the 24-cell ==== Another useful set of coordinates for the 24-cell is comprised solely of orthogonal great squares. In this coordinate system the 24-cell lies cell-up, and the great squares are aligned with the squares of the coordinate lattice. {| class="wikitable" !Great squares of the 24-cell<BR>Cartesian{{s|2}}(<small>±{{radic|{{sfrac|1|2}}}}, ±{{radic|{{sfrac|1|2}}}}, 0, 0</small>) |- | {| class="wikitable" !Hopf{{s|2}}({<4}{{sfrac|𝜋|2}}, {{sfrac|𝜋|4}}, {<4}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" !(𝜉_''i'', {{sfrac|𝜋|4}}, 𝜉_''j'') !0{{sfrac|𝜋|2}} !1{{sfrac|𝜋|2}} !2{{sfrac|𝜋|2}} !3{{sfrac|𝜋|2}} !sin !cos |- !0{{sfrac|𝜋|2}} |(<small> {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small> {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%| 1}} |- !1{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%| 1}} !{{font|size=75%| 0}} |- !2{{sfrac|𝜋|2}} |(<small>-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}},-0, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0, {{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}, 0</small>) |(<small>-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}},-0, 0</small>) !{{font|size=75%| 0}} !{{font|size=75%|-1}} |- !3{{sfrac|𝜋|2}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0, 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}</small>) |(<small> 0,-{{radic|{{sfrac|1|2}}}}, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-1}} !{{font|size=75%| 0}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|−1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|1|2}}}}}} |} |} |- | {| class="wikitable" !Hopf{{s|2}}({1 3 5 7}{{sfrac|𝜋|4}}, {≤1}{{sfrac|𝜋|2}}, {1 3 5 7}{{sfrac|𝜋|4}})₄ |- | {| class="wikitable" !(𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> 0, {{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}}, {{radic|{{sfrac|1|2}}}}, 0</small>)}} |{{font|color=gray|(<small> 0,-{{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>)}} |(<small> 0, {{radic|{{sfrac|1|2}}}},-{{radic|{{sfrac|1|2}}}}, 0</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" !(𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'') !1{{sfrac|𝜋|4}} !3{{sfrac|𝜋|4}} !5{{sfrac|𝜋|4}} !7{{sfrac|𝜋|4}} !sin !cos |- !1{{sfrac|𝜋|4}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !3{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0, {{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !5{{sfrac|𝜋|4}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) |{{font|color=gray|(<small>-{{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} |- !7{{sfrac|𝜋|4}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |{{font|color=gray|(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>)}} |(<small> {{radic|{{sfrac|1|2}}}}, 0, 0,-{{radic|{{sfrac|1|2}}}}</small>) !{{font|size=75%|-{{sfrac|{{radic|2}}}}}} !{{font|size=75%| {{sfrac|{{radic|2}}}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|sin 𝜂 {{=}} 1}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%|−{{radic|{{sfrac|1|2}}}}}}||{{font|size=75%| {{radic|{{sfrac|1|2}}}}}}|| ||{{font|size=75%|cos 𝜂 {{=}} 0}} |} |} |} ==== Great squares and rectangles of the compound of dual 24-cells ==== Two sets of coordinates for the 24-cell have now been given (above). In the first (great squares and rectangles) the 24-cell lies vertex-up, and in the second (great squares only) it lies cell-up. The 24-cell being a self-dual 4-polytope, these two 24-cells are duals of each other, the vertices of one lying at the cell centers of the other, and the union of their two sets of coordinates is a 48-vertex compound of duals of the same radius. In this compound of two 24-cells, 24 16-cells are inscribed: the 3 inscribed in each of the dual 24-cells, and 18 others which span the two 24-cells.{{Sfn|Waegell|Aravind|2009}} Each of these 24 16-cells, with its 4 orthogonal axes and 6 orthogonal planes, constitutes an equivalent ''basis'' for a Cartesian coordinate system, and contains three pairs of completely orthogonal planes, each pair of which intersects all 8 of the 16-cell's vertices. The compound of two 24-cells has 24 axes and 24 bases, with each basis consisting of four axes and each axis occurring in four bases. The 3 16-cells inscribed in each 24-cell are disjoint from each other and the dual 24-cell's 3 inscribed 16-cells. Each 24-cell has 18 central squares, and the 18 spanning 16-cells are each comprised of 4 vertices comprising a great square from one of the 24-cells, and another 4 vertices comprising a great square from the dual 24-cell.{{Efn|In each of the 18 16-cells, the two central squares from dual 24-cells are orthogonal, but not completely orthogonal. Each is already completely orthogonal to another central square within the same 24-cell (within the same 16-cell), and in 4-space a plane cannot be completely orthogonal to more than one other plane through the same point. Although their two sets of 4 vertices are disjoint, that is not because their square planes are completely orthogonal; rather, their planes intersect in a line, but their vertices remain disjoint because the line of intersection does not pass through any of their vertices.}} ==== Great hexagons and squares of the compound of dual 24-cells ==== In a single 24-cell, the hexagonal central planes lie at 60 degrees to each other and to the square central planes. The central planes orthogonal to the hexagonal planes are digons: they intersect only 2 vertices. Consequently it is not possible to find Hopf coordinates for the 24-cell in which both the 𝜉_''i'' and 𝜉_''j'' orthogonal invariant planes contain hexagons. However, the 24-cell and its unscaled dual form a compound in which the dual 24-cells are separated by a Clifford displacement (an isoclinic rotation) of 30 degrees. The hexagonal planes are still not orthogonal to each other in this compound (they are inclined at 30 degrees or 60 degrees to each other), so it is still impossible to find a 6 x 6 array of Hopf coordinates, but the hexagonal planes of one 24-cell are orthogonal to the square planes of the other. In this compound of 48 vertices, a hexagonal wz plane and a square xy plane can be the invariant planes of a 6 x 4 array of Hopf coordinates. {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the compound of dual 24-cells Cartesian{{s|3}}(<small>±_, ±_, _, _</small>)<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {1 2}{{sfrac|𝜋|6}}, {<small><4</small>}{{sfrac|𝜋|2}})₁ |- | {| class="wikitable" style="white-space:nowrap;" !|(𝜉_''i'', 1{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}}, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}},-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0,-{{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}}, 0) |( 0, {{radic|{{sfrac|3|16}}}},-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|1|2}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 2{{sfrac|𝜋|6}}, 𝜉_''j'')||0{{sfrac|𝜋|2}}||1{{sfrac|𝜋|2}}||2{{sfrac|𝜋|2}}||3{{sfrac|𝜋|2}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%| 0}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%|sin 𝜂 {{=}} {{radic|{{sfrac|3|4}}}}}}|| |- !{{font|size=75%|cos}}||{{font|size=75%| 1}}||{{font|size=75%| 0}}||{{font|size=75%|-1}}||{{font|size=75%| 0}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|1|2}}}} |} |} ==== Great hexagons and digons of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and digons of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤1}{{sfrac|𝜋|2}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 0, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜋, 𝜉_''j'')||0||𝜋||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( _, _, _, _) |( _, _, _, _) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|-1}}|| ||{{font|size=75%|cos 𝜼 {{=}} -1}} |} |} ==== Great hexagons and squares of the 24-cell ==== {| class="wikitable" style="white-space:nowrap;" !Great hexagons and squares of the 24-cell Cartesian (<small>±_, ±_, _, _</small>)<BR> Hopf ({<small><6</small>}{{sfrac|𝜋|3}}, {≤3}{{sfrac|𝜋|6}}, {<small><2</small>}𝜋)₁ |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 0𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( 0, 1, 0, 0) |( 0, {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}) |( 0, {{sfrac|1|2}}, 0, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, 0, 0,-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 1, {{sfrac|3|4}}, 0) |( {{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|3|4}}, {{sfrac|1|2}}) |( {{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}}, {{sfrac|3|4}}, 0) |( 0, 0, {{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( 1, 1, {{radic|{{sfrac|3|4}}}}, 0) |( {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}) |( {{sfrac|1|2}}, {{sfrac|1|2}}, {{sfrac|1|2}}, 0) |( 0, 0, {{radic|{{sfrac|3|4}}}},-{{sfrac|1|2}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |(-{{radic|{{sfrac|3|4}}}}, 1,-{{sfrac|3|4}}, 0) |(-{{sfrac|3|4}}, {{radic|{{sfrac|3|4}}}},-{{sfrac|3|4}}, {{sfrac|1|2}}) |(-{{radic|{{sfrac|3|16}}}}, {{sfrac|1|2}},-{{sfrac|3|4}}, 0) |( 0, 0,-{{sfrac|3|4}},-{{sfrac|1|2}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !(𝜉_''i'', 𝜂, 1𝜋)||0{{sfrac|𝜋|6}}||1{{sfrac|𝜋|6}}||2{{sfrac|𝜋|6}}||3{{sfrac|𝜋|6}}|| {{font|size=75%|sin}}||{{font|size=75%|cos}} |- !0{{sfrac|𝜋|3}} |( a, , 0, 0) |( 0, {{sfrac|1|2}}, 0, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|2}}, 0, 0) |( 0, {{sfrac|1|2}}, 0,-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 0}}||{{font|size=75%| 1}} |- !1{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !2{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}}, {{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| {{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !3{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, 0, {{sfrac|1|2}}, 0) |( 0, 0, {{sfrac|1|2}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%| 1}}||{{font|size=75%| 0}} |- !4{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}},-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0,-{{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|-{{sfrac|1|2}}}} |- !5{{sfrac|𝜋|3}} |( {{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, {{radic|{{sfrac|3|4}}}}) |(-{{radic|{{sfrac|3|4}}}}, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}}, 0) |( 0, {{sfrac|1|4}},-{{radic|{{sfrac|3|16}}}},-{{radic|{{sfrac|3|4}}}}) !{{font|size=75%|-{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%| {{sfrac|1|2}}}} |- !{{font|size=75%|sin}}||{{font|size=75%|0}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|1}}||{{font|size=75%|sin 𝜉_''j'' {{=}} 0}}|| |- !{{font|size=75%|cos}}||{{font|size=75%|1}}||{{font|size=75%|{{radic|{{sfrac|3|4}}}}}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|0}}|| ||{{font|size=75%|cos 𝜉_''j'' {{=}} -1}} |} |} ==== Dual fibrations ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[fiber bundle]] of Clifford parallel great circles is a discrete [[Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once. {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell<BR> Hopf ({<<small>6</small>}{{sfrac|𝜋|3}}, {<small><10</small>}{{sfrac|𝜋|20}}, {<small><2</small>}𝜋)₁<BR> Cartesian ({<small>0, ±1, 0, 0</small>}) (<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>) ([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle pentagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle pentagons of the 600-cell<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>])<BR> Hopf{{s|3}}({<small>0 2 4 6 8</small>}{{sfrac|𝜋|5}}, {<small><24</small>}{{sfrac|𝜋|48}}), {<small>1 3 5 7 9</small>}{{sfrac|𝜋|5}})₅ |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'',{{sfrac|𝜋|4}}, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} {{sfrac|{{radic|2}}|2}}}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|(<small>𝜉_''i'', 0, 𝜉_''j''</small>) !1{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|2{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|6{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0)) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|8{{sfrac|𝜋|5}} |( 0, {{sfrac|ϕ|2}}, a, 0)||( 0, {{sfrac|ϕ⁻¹|2}}, b, 0)||( 0, -1, 0, 0)||( 0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||( 0,{{sfrac|ϕ|2}}, -a, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|{{font|size=75%|sin}} !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|0}}||{{font|size=75%|-b ≈ -0.951}}||{{font|size=75%|-a ≈ -0.588}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2|{{font|size=75%|cos}} !{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|-1}}||{{font|size=75%|{{sfrac|ϕ^-1|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle squares and hexagons of the 120-cell==== {| class="wikitable" !colspan=1|Great circles of the 120-cell:<BR> Hopf{{s|3}}({<small><6</small>}{{sfrac|𝜋|3}}, {<small>≤24</small>}{{sfrac|𝜋|48}}, {<small><4</small>}{{sfrac|𝜋|2}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]){{s|3}}... |- | |} ====Great circle decagons and hexagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons and hexagons of the 600-cell:<BR> Hopf{{s|3}}({<10}{{sfrac|𝜋|5}}, {<small>≤1</small>}{{sfrac|𝜋|2}}, {<<small>6</small>}{{sfrac|𝜋|3}})₁<BR> Cartesian{{s|3}}({<small>0, ±1, 0, 0</small>}){{s|3}}(<small>±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|1|2}}</small>){{s|3}}([<small>±{{Sfrac|φ|2}}, ±{{Sfrac|1|2}}, ±{{Sfrac|φ⁻¹|2}}, 0</small>]) |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 0{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|𝜉_''i'', 1{{sfrac|𝜋|2}}, 𝜉_''j'' !0{{sfrac|𝜋|3}}||1{{sfrac|𝜋|3}}||2{{sfrac|𝜋|3}}||3{{sfrac|𝜋|3}}||4{{sfrac|𝜋|3}}||5{{sfrac|𝜋|3}}||{{font|size=75%|sin}}||{{font|size=75%|cos}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, 1, 0, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|1}} |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}} |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|0}}||{{font|size=75%|−1}} |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}} |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}} |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}} |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, a, {{sfrac|ϕ|2}}, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0) !{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|sin}} !{{font|size=75%|0}}||{{font|size=75%|{{sfrac|{{radic|3}}|2}} ≈ 0.866}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|sin 𝜂 {{=}} 0}}|| |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|cos}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|1|2}}}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}|| ||{{font|size=75%|cos 𝜂 {{=}} 1}} |} |} ====Great circle decagons of the 600-cell==== {| class="wikitable" !colspan=1|Great circle decagons of the 600-cell:<BR> Hopf{{s|3}}({<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}}, {<small>0 1 2 3 4 5</small>}{{sfrac|𝜋|10}}, {<small>0 1 2 3 4 5 6 7 8 9</small>}{{sfrac|𝜋|5}})₅<BR> Cartesian{{s|3}}... |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 0{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0))||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0))||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |(0, 1, 0, 0)||(0, {{sfrac|ϕ|2}}, a, 0)||(0,{{sfrac|ϕ⁻¹|2}}, b, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, b, 0)||(0, -{{sfrac|ϕ|2}}, a, 0)||(0, −1, 0, 0)||(0, -{{sfrac|ϕ|2}}, -a, 0)||(0,-{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0,{{sfrac|ϕ⁻¹|2}}, -b, 0)||(0, {{sfrac|ϕ|2}}, -a, 0) |- !style="white-space:nowrap;"|{{font|size=75%|sin}}||{{font|size=75%|𝜂 {{=}} 0}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !style="white-space:nowrap;"|{{font|size=75%|cos}}||{{font|size=75%|𝜂 {{=}} 1}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 1{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}},{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|bϕ^-1|2}}) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², {{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, {{sfrac|aϕ^-1|2}}) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, 0)||(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ|2}}, -ab, 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -{{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ^-1|2}}, -b², 0) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, {{sfrac|bϕ|2}}, -ab, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}},-{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|(-{{sfrac|ϕ^-1|2}}, -b, 0, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|bϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|ϕ^-2|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|bϕ^-1|2}}) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}},-{{sfrac|1|4}}, ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -b, 0, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, -{{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ^-1|2}}, -b², -{{sfrac|aϕ^-1|2}}) |style="white-space:nowrap;" align=center|({{sfrac|1|4}}, {{sfrac|bϕ|2}}, -ab, -{{sfrac|aϕ^-1|2}}) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ^-1|2}} ≈ 0.309}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} b ≈ 0.951}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} {{s|5}}<small>{{sfrac|bϕ|2}} ≈ 0.769{{s|5}}ab ≈ 0.559{{s|5}}{{sfrac|bϕ^-1|2}} ≈ 0.294{{s|5}}b² ≈ 0.905</small> |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 2{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 3{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 4{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |- | {| class="wikitable" !colspan=2 style="white-space:nowrap;"|(𝜉_''i'', 5{{sfrac|𝜋|10}}, 𝜉_''j'') !0{{sfrac|𝜋|5}}||1{{sfrac|𝜋|5}}||2{{sfrac|𝜋|5}}||3{{sfrac|𝜋|5}}||4{{sfrac|𝜋|5}}||5{{sfrac|𝜋|5}}||6{{sfrac|𝜋|5}}||7{{sfrac|𝜋|5}}||8{{sfrac|𝜋|5}}||9{{sfrac|𝜋|5}} |- !colspan=2|0{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|1{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|2{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|3{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|4{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|5{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|6{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|7{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|8{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2|9{{sfrac|𝜋|5}} |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |style="white-space:nowrap;" align=center|(0, 0, 0, 0) |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} a ≈ 0.588}} !{{font|size=75%|0}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|b ≈ 0.951}}||{{font|size=75%|a ≈ 0.588}}||{{font|size=75%|0}}||{{font|size=75%|-a ≈ −0.588}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-b ≈ −0.951}}||{{font|size=75%|-a ≈ −0.588}} |- !colspan=2 style="white-space:nowrap;"|{{font|size=75%|𝜂 {{=}} {{sfrac|ϕ|2}} ≈ 0.809}} !{{font|size=75%|1}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|−{{sfrac|ϕ^-1|2}} ≈ −0.309}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−1}}||{{font|size=75%|−{{sfrac|ϕ|2}} ≈ −0.809}}||{{font|size=75%|−{{sfrac|ϕ⁻¹|2}} ≈ −0.309}}||{{font|size=75%|{{sfrac|ϕ⁻¹|2}} ≈ 0.309}}||{{font|size=75%|{{sfrac|ϕ|2}} ≈ 0.809}} |} |} == Equilateral rings == Equilateral rings are those which can be constructed out of equilateral triangles on the circumference of a sphere. === Borromean equilateral rings === The {1,1,1} torus knot. The vertices of the regular icosahedron form five sets of three concentric, mutually [[orthogonal]] [[golden rectangle]]s, whose edges form [[Borromean rings]]. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures <math>2\times 4</math>. These three rectangles are the shortest possible representation of the Borromean rings using only edges of the [[integer lattice]]. ... == Kinematics == In 3D we have the kinematic transformations of the cuboctahedron (cuboctahedron, icosahedron, jessen's, golden icosa?, octahedron-2, tetrahedron-4?) and their duals, the transformations of the dodecahedron: two sets of nesting Russian dolls (or perhaps one set?). In 4D we apparently have instances of the cuboctahedron nestings in the 600-cell (and perhaps the dodecahedron nestings as well, in the 120-cell?). This suggests that the unit-radius sequence of 4-polytopes may contain dynamic as well as static nestings. From [[W:Kinematics of the cuboctahedron#Duality of the rigid-edge and elastic-edge transformations|Kinematics of the cuboctahedron § Duality of the rigid-edge and elastic-edge transformations]]: <blockquote> Finally, both transformations are pure abstractions, the two limit cases of an infinite family of cuboctahedron transformations in which there are two elasticity parameters and no requirement that one of them be 0. ... In engineering practice, only a tiny amount of elasticity is required to allow a significant degree of motion, so most tensegrity structures are constructed to be "drum-tight" using nearly inelastic struts ''and'' cables. A '''tensegrity icosahedron transformation''' is a kinematic cuboctahedron transformation with reciprocal small elasticity parameters.</blockquote> From [[W:24-cell#Double rotations|24-cell § Double rotations]]: <blockquote>The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes of rotation at once.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. </blockquote> This suggests that the reciprocal limit-case kinematic transformations (with one orthogonal elasticity parameter or the other equal to 0) may be expressable as double rotations, considering the relativity of such transformations.{{Efn|name=transformations}} === Completely orthogonal planes === In three dimensions (on polyhedra) there are no disjoint great circles. Every pair of great circles intersects at two points, the endpoints of a diameter of the sphere. But in four dimensions (on polychora) every great circle is disjoint from exactly one other great circle: the one to which it is completely orthogonal. If we consider the two polyhedral great circles' common diameter to be a common axis of rotation, we can see that rotating either circle about that axis generates the whole polyhedron; thus either circle by itself can generate the whole polyhedron by rotation. But in four dimensions two completely orthogonal great circles have no common axis of rotation (no points at all in common, all their points are disjoint). Clearly either circle by itself cannot generate the entire polychoron by rotation about a fixed axis. Rotating each circle about an axis generates only half the points on the 3-sphere - rotating the other circle generates the other half of them. Rotation about a fixed axis in four dimensions necessarily leaves an entire plane fixed, and generates only a 3-dimensional polyhedron. If the great circle in the xy plane is rotated about the y axis, only a 2-sphere is generated, and all the points on the 3-sphere outside the hyperplane w = 0 will be left out. In the 24-cell and the 8-cell, which are radially equilateral, ... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. {{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Coxeter mirrors ==== [[W:Coxeter group|Coxeter group]] theory (the mathematics of [[W:Polytopes|polytopes]] of any number of dimensions) can be informally described as a ''finite closed system of mirrors'', or the [[W:Geometry|geometry]] of multiple [[W:Mirror#Mirror images|mirror images]]. Mathematically it is the equivalent of the theory of finite [[W:Reflection group|reflection groups]] and [[W:Root system|root systems]], expressed in a different mathematical language. But unlike those [[W:Group theory|group theory]] languages, its principal objects can be defined in the most intuitive and elementary way: <blockquote> Imagine a few (semi-transparent) mirrors in ordinary three dimensional space. Mirrors (more precisely, their images) multiply by reflecting in each other, like in a [[W:Kaleidoscope|kaleidoscope]] or a [[W:Gallery of mirrors|gallery of mirrors]]. A ''closed system of mirrors'' is what we see when we look into such a kaleidoscope.{{Sfn|Borovik|2006|loc=§1. Mirrors and Reflections|pp=18-19}}</blockquote> Coxeter refers to the space between two parallel mirrors as "the region of possible objects". This suggests that space itself is being generated by the objects that it contains, which parallel reflections multiply infinitely. When the mirrors are not parallel, the multiplication may be finite rather than infinite (provided the dihedral angle between the mirrors is a submultiple of 𝝅). The space between two or more such ''intersecting'' mirrors is called the fundamental region, and it constitutes a proto-space of a finite number of dimensions with mirrors as its bounding walls. Coxeter says "the lines of symmetry or circles of symmetry or planes of symmetry are mirrors reflecting the whole pattern into itself. We count these circles of symmetry by counting their pairs of antipodal points of intersection with a single equator."{{Sfn|Coxeter|1938}} He characterizes great circles of symmetry in terms of the Petrie ''h''-gon i.e. the 3''h''/2 circles of symmetry possessed by each Platonic solid. For example the Petrie polygon of the octahedron and the cube is the hexagon, so they have 18/2 = 9 great circle planes of symmetry (mirrors). Each is generated by placing a single point-object in the fundamental region (off the surface of any mirrors); reflected in the mirrors it multiplies into all the vertices of the polytope. ==== Translation-rotations ==== An object displacement in space may be a rotation (which leaves at least one point invariant), a translation (which does not), or a combination of both.{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations|pp=33-38}} The circular path of a rotation may be combined with a translation in the axial direction of the rotation yielding a ''screw-displacement'', the general case of a displacement.{{Sfn|Coxeter|1973|loc=§3.14|ps=; "''Every displacement is a screw-displacement'' (including, in particular, a rotation or a translation)."|p=38}} In three dimensions, a screw-displacement is a simple helix, as its name suggests. In four dimensions, the circular path of an isoclinic rotation is already a helix (a geodesic isocline), and there are four orthogonal axial directions of rotation.{{Sfn|Coxeter|1973|loc=§12.1 Orthogonal transformations|pp=213-217|ps=; "The general displacement preserving the origin in four dimensions is a ''double rotation''.... The two completely orthogonal planes of rotation are uniquely determined except when 𝜉₂ {{=}} 𝜉₁, in which case... we have a ''Clifford displacement''."}} In the unit-radius 24-cell an isoclinic rotation by 60° moves each vertex {{radic|3/4}} ≈ 0.866 in each of four orthogonal directions at once, a total Pythagorean distance as if it had moved straight along a combined {{radic|3}} ≈ 1.732 chord. When the rotation is combined with a unit translation, the {{radic|1}} translation vector must be divided among all four rotation vectors. The vertex moves {{radic|3/4 + 1/4}} = {{radic|1}} in each of the four orthogonal directions, moving a combined Pythagorean distance of {{radic|4}}, the maximum ''unit displacement'' in 4 dimensions: the distance which is the [[W:Tesseract#Radial equilateral symmetry|long diameter of the 4-hypercube (tesseract)]]. This movement ''could'' take the vertex to its antipodal vertex {{radic|4}} away, if the direction of the translation is so aligned, but in all other cases it will take it to a point outside the 24-cell. ==== Relative screw displacement ==== (screw displacement) QT = QR² = R⁴ = Q² (double rotation){{Sfn|Coxeter|1973|P=217|loc=§12.2 Congruent transformations}} A screw displacement in four dimensions is equivalent to a double rotation, by the principle of relativity. There are only two kinds of screw displacements possible in only ''four'' dimensions: the single kind (which also occurs in three dimensions), and the double kind (which requires four dimensions). The latter kind of screw displacement is inherently double, the product of 4 reflections, just like a double rotation. In fact it ''is'' just a double rotation, seen from a moving inertial reference frame. A product of two reflections is a (simple) rotation, unless the reflecting facets are exactly parallel, in which case it is a translation. In other words, a translation is just a rotation on a circle of infinitely long radius (a straight line). A screw displacement is just a double rotation in which one rotation (the one which is the translation) has infinitely long radius, i.e. a vanishingly small angle of rotation (near 0 degrees) compared to the angle of the other rotation (between 0 and 90 degrees). The screw displacement looks like a simple rotation within a three dimensional reference frame that is in uniform translation on the 4th dimension axis at near-infinite velocity. In the case of actual moving objects, no actual translation is an infinite straight line, and no velocity is infinite; any moving object that describes a screw-displacement is presumably moving on a curved translation under the influence, at least, of some distant gravitational force, however miniscule, and the radius of the translation-rotation it describes is merely very long, not infinite. So we can say that there can be only one range of situations in actuality, no perfectly straight translations but only double rotations of more or less eccentricity, which will appear to be simple rotations inside a 3-dimensional reference frame moving uniformly along the 4th dimension "translation" axis. Its eccentricity is merely a matter of choice of reference frame: it looks like a simple rotation in a reference frame moving uniformly with the translation, and a double rotation (of perhaps extreme eccentricity) in a reference frame that is not moving with the translation. Within this range of possibilities, only one possibility is ''not'' eccentric: the case of the equi-angled double rotation, called an isoclinic rotation or Clifford displacement. Since the ratio of eccentricity is a matter of choice of reference frame, we may adopt as our preferred reference frame (of any actual screw displacement occuring in practice) the inertial reference frame in which the double rotation is isoclinic. ==== Total internal reflection ==== The phenomenon in physics known as [[W:Total internal reflection|total internal reflection]] keeps light confined within one strand of a [[W:Optical fiber|fiber optic cable]]. Isoclinic rotations and screw displacements in 4-dimensional space are both the consequence of four symmetrical reflections, and their propagation corresponds to a total internal reflection within the 3-sphere. Consequently 4-dimensional space itself acts as a [[W:waveguide|waveguide]] for isoclinically rotating objects during their translation. This provides a purely geometric model for the [[w:inertia|inertia]] of mass-carrying objects, and for light-wave propagation. ==== Isoclines ==== In an isoclinic rotation the vertices of a 4-polytope such as the 24-cell move on ''isoclines'', which are helical circles that wind through all four dimensions. Isoclinic rotations are [[W:chiral|chiral]], occuring in left-handed and right-handed mirror-image pairs in which the moving vertices reach different destinations along left or right paths. The isoclines themselves however (the helical paths of the moving vertices) are not chiral objects: they are non-twisted (directly congruent) ''circles'', of a special 4-dimensional kind. Every left isocline path in a left-handed rotation acts also as a right isocline path in some right-handed rotation, in some ''other'' left-right pair of isoclinic rotations. Isoclinic rotations and their isoclines occur as fibrations (fiber bundles of non-intersecting but interlinked circles), with each fibration consisting of a single distinct left-right pair of isoclinic rotations. Each distinct left (or right) rotation has some number of isoclines, which are the circular paths along which its vertices orbit, with each vertex confined to a single isocline circle throughout the rotation. The multiple isoclines of a distinct left or right rotation do not intersect each other; they are Clifford parallel, which means that they are curved lines which are parallel to each other, in the sense that they are the same distance apart at all of their corresponding (nearest) points. Thus the moving vertices in an isoclinic rotation circulate in parallel disjoint sets. In the 24-cell's characteristic kind of isoclinic rotation, the moving vertices circulate on skew hexagon isoclines, in 4 parallel disjoint sets of 6 moving vertices each. This characteristic kind of isoclinic rotation occurs in four different fibrations: there are four distinct left-right rotation pairs. In each distinct left (or right) rotation, there are 4 Clifford parallel isoclines, each of which is a helical circle through 6 vertex positions. The 4 disjoint circles of 6 vertices pass through all 24 vertices of the 24-cell, just once. Although the four isocline circles do not intersect, they do pass through each other as do the links of a chain, but unlike linked circles in three-dimensional space, they all share the same center point. ==== Polygrams and cell rings ==== The isoclines of [[W:24-cell#Isoclinic rotations|24-cell isoclinic rotations]] in ''hexagonal'' central planes have 6 chords which form a [[W:Skew polygon|skew]] [[W:hexagram|hexagram]]. Every [[W:24-cell#Helical hexagrams and their isoclines|hexagram isocline]] is contained within the volume of a distinct [[W:24-cell#Cell rings|ring of 6 face-bonded octahedral cells]] which, like its axial great circle hexa''gon'' is an equatorial [[W:24-cell#Rings|ring of the 24-cell]]. Each 6-cell ring contains the left and right isoclines of a distinct left-right pair of isoclinic rotations. [[W:24-cell#6-cell rings|The 6-cell ring itself]] is not a chiral object because it contains ''both'' mirror-image isoclines: they are twisted in opposite directions (around each other), but the 6-cell ring that contains them both has no [[W:Torsion of a curve|torsion]]. ==== Reflections ==== Because each octahedral cell volume can be subdivided into 48 orthoschemes (the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the octahedron]] and [[W:24-cell#Characteristic orthoscheme|of the 24-cell]]), we can be more precise in describing the cell ring each isocline stays within. Within the 6-cell ring of face-bonded spherical octahedra is a ring of face-bonded spherical characteristic tetrahedra that contains the isocline. Several characteristics of this ring are evident from the nature of [[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoschemes]]: it consists of alternating left-hand and right-hand orthoschemes (mirror images of each other in their common face), and each of them contains one and only one vertex of the 24-cell.{{Efn|In the 24-cell 48 [[W:24-cell#Characteristic orthoscheme|characteristic 4-orthoschemes (5-cells)]] meet at each of the 24 vertices, and 48 [[W:Octahedron#Characteristic orthoscheme|characteristic 3-orthoschemes (tetrahedra)]] fill each of the 24 octahedral cells. The number of characteristic orthoschemes in a regular polytope is the ''order'' of its [[W:Coxeter group|symmetry group]], which for the octahedron is 48, and for the 24-cell is 1152 {{=}} 24 * 48.}} The sequence of orthoscheme vertices is the same as [[W:24-cell#Helical hexagrams and their isoclines|the sequence of isocline vertices]], except that it includes the vertices near-missed by the isocline as well as those the isocline intersects (12 distinct vertices instead of 6), and the orthoscheme sequence will have many more than 12 elements because it includes more than two orthoschemes incident to each vertex.{{Efn|The ring contains more orthoschemes than necessary to contain the isocline, because of our stipulation that all pairs of adjacent orthoschemes be face-bonded. The isocline intersects only a vertex, or only an edge, of some orthoschemes in the face-bonded ring. At each vertex it hits, the isocline passes between two orthoschemes that touch only at that vertex. Near each vertex that it misses, the isocline passes through an edge between two orthoschemes that touch only at that edge.}} === Physical space === We attempt to be more precise about the shape of this 4-space, and in particular, the cause of its shape, i.e. the relationship between the fundamental forces observed in nature and this spatial geometry. As Einstein did in his 1923 thought experiment, we identify the observed 3-dimensional cosmos (everything in it up to some large scale such as a galaxy) as a thin manifold embedded in a Euclidean (i.e. flat) 4-dimensional space of the kind elucidated by Coxeter. Further we postulate that every mass-carrying particle in this space is in motion at speed <math>c</math> relative to the 4-dimensional space itself. The 4-space therefore has a quasi-ether-like existence insofar as it defines a field at absolute rest, relative to which the motion of all particles at speed <math>c</math> can be universally compared, with the important provision that no particle, anywhere, is ever at absolute rest itself with respect to this field. The condition of absolute rest is an abstract condition attributable only to the field, and never to any tangible object. Thus the field itself (4-dimensional Euclidean space) is an abstraction somewhat more tangible than Mach's relative space, but much less tangible than the luminous ether, much as Einstein found 4-dimensional spacetime to be. Directions and distances can be fixed universally within the Euclidean 4-space field (they are invariant for all observers regardless of their direction of motion within the field), but locations can only be relative to some object (not to the field itself), and all 4-dimensional velocities are invariant: they are always <math>c</math> with respect to the field, for any mass-bearing particle or observer. Einstein's general relativity identifies gravity as a fictitious force, attributable to the shape of the 3-dimensional manifold rather than to an attractive force acting instantaneously at a distance. The 3-dimensional manifold is said to be singular and universal (all objects in the universe lie within it), but its shape varies by location. It is assumed to curve or dimple in the vicinity of massive objects, such that other objects fall into the dimples naturally in the course of following straight-line paths (geodesics) through it. In general relativity 3-dimensional space is flat near each observer, but there is no universally flat space except in regions far removed from massive objects, i.e. in places where the simplifications of the theory of special relativity can be assumed. But in Euclidean relativity this 3-dimensional manifold is embedded in a 4-dimensional Euclidean space, and that 4-space field is flat universally, at all times for all observers. Furthermore, we only assume that the 4-dimensional space is singular and universal; there might be more than one 3-dimensional manifold embedded within it, and the 3-manifolds do not necessarily intersect. In Euclidean relativity we expect that not just gravity, but all the fundamental forces observed in nature, are an expression of the local geometry of a 3-space manifold embedded in Euclidean 4-space. By ''expression'' we mean the consequence of a transformation such as a reflection, rotation or translation, i.e. operations of the fundamental Coxeter symmetry groups, which characterize the Euclidean geometry of the universal space in which the 3-space manifolds are embedded. === Closed 3-manifolds embedded in 4-space === The only reason to suppose there is only one such closed, curved soap-bubble 3-manifold in our 4-space universe is the assumption that every particle in the universe had a common origin, at a single point in 4-space and a single moment in time in a big bang, and even in that case there could be many such soap-bubble 3-manifolds in existence now. One can certainly model the observed universe as a single closed, curved 3-manifold, and cosmologists do, but there is no more proof that this model is the correct one than there was for the model with the earth at the center of the universe. Whether we determine that light propagates through 4-space in straight lines, or only on geodesic curves along 3-manifolds, we can only determine ''by looking'' that the space ''near'' us is resolutely 3-dimensional (not admitting the construction of four mutually perpendicular axes, only three). When we look out very far, at distant galaxy clusters for example, we have no way of determining whether we are looking through three dimensional space or four dimensional space. All those distant objects we see ''might'' lie in the same 3-manifold (perhaps on the same rough 3-sphere) that we do, but why should they have to? Might they not lie on separate 3-sphere soap bubbles, vastly distant from ours, whether or not all the soap bubbles had a common origin at one place and time? When we consider the ways in which particles propagating at the speed of light might reach us, considering that we ourselves are formations of particles propagating at the speed of light (all together in almost the same direction), it is clear that we ought not to expect to be overtaken by such particles emanating from stars on the opposite side of our own 3-manifold (from our antipodes, so to speak), because even such a particle redirected exactly backwards along the proper time axis of a star at our antipodes could only follow us along our opposite-direction proper-time axis at a fixed distance forever, never overtaking us, as we travelled in the same direction through 4-space at the same speed <math>c</math> forever. Or, since our own path through 4-space is a helical one (as we are engaged in numerous concentric orbits), if the pursuing particle's path through 4-space were a straighter one, it might in principle overtake us eventually, but probably not in our actual experience, and never as a particle moving relative to us at nearly the speed <math>c</math>. Therefore we should not expect to receive such particle radiation from the backside stars of our own 3-manifold. === The speed of light === So far, however, these considerations can apply only to mass particle radiation, not to light signals, since we have not yet described how light particles (photons) propagate through 4-space. We have suggested that elementary rigid objects propagate themselves by discrete Coxeter transformations,{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} </blockquote>|name=transformations}} that atomic mass particles are elementary rigid objects in some sense, and that all such particles are transforming at a constant rate <math>c</math> in various directions through Euclidean 4-space. But so far, the motion of mass particles is the only kind of motion we have described; we have not given an account of the nature of light signals, or the manner of their propagation, except to observe that light signals propagate through 4-space ''faster'' than <math>c</math>, as a consequence of the fact that they are observed to propagate through ''3-space'' at speed <math>c</math>. How much faster than <math>c</math> must photons be traveling through 4-space, since they appear to be traveling at <math>c</math> relative to two observers: one in the reference frame of the electron emitting the photon, and one in the reference frame of the electron absorbing the photon? If both observers are themselves traveling through 4-space at speed <math>c</math> as we have stipulated, then even in the case where their direction through 4-space is the same (they are at rest with respect to each other in the same reference frame), a photon that passes between them must travel at speed <math>\sqrt{2}c</math> if it makes the trip in a straight line, or even faster if it zig-zags in some fashion. Fortunately, this requirement is not at all paradoxical, since in a system of particles translating themselves through 4-space at the rate of <math>c</math> transforms per unit time, some things ''do'' move faster than speed <math>c</math>. A transforming rigid object with a translational motion at rate <math>c</math> may simultaneously have an orthogonal rotational motion at rate <math>c</math>, such that its component parts (e.g. each vertex of a rotating-translating 4-polytope) may displace themselves in 4-space ''more'' than one object-diameter in each discrete transformation; the combined rotating-and-translating velocity through 4-space of a ''component'' may be as much as twice the translational velocity of the whole rigid object, <math>\sqrt{4}c</math> rather than <math>\sqrt{1}c</math>, the [[W:8-cell#Radial equilateral symmetry|diagonal of an atomic unit 4-cube]] rather than its edge length. But the component points of such a rotating rigid object are all traveling in different directions at any instant, and the combined motion of the object as a whole cannot be other than <math>c</math>. Therefore a propagating light signal (a photon) is not a rigid atomic object, but some propagation of one of its component parts. Of course this agrees perfectly well with our understanding of photons as emissions of electrons, even if electrons are themselves rigid atomic objects translating themselves through 4-space at the rate of <math>c</math> transforms per unit time. The only "paradox" is linguistic in nature: <math>c</math> is not the "speed of light", it is the speed of matter (all mass-carrying particles) through 4-space. The actual ''speed of photons'' through 4-space is <math>2c</math>, as opposed to their observed speed through 3-space of <math>c</math>. === ... === A light signal (photon) propagates at speed c relative to either the emitting or the absorbing inertial reference frame (which reference frames are themselves in motion at speed c relative to their common 4-space stationary reference frame, but perhaps in different directions through that 4-space). ==== How soap bubble 3-manifolds behave in 4-space ==== 120 similar 2-sphere soap bubbles (spherical dodecahedra) tiling the 3-sphere, meeting at 120 degrees three-around each edge, and four at each vertex.{{Sfn|Stillwell|2001|p=24|loc=Figure 7. Soap bubble 120-cell}} === Atmospheric 3-membrane === What if the 3-space we observe (the visible universe) were filled with a gas, contained in some manner within the (thin) 3-membrane? Like the (thin) atmosphere of the earth (in a 3-dimensional analogy). In fact it is gas-filled, if the 4-space inside and outside the 3-membrane is empty. That difference is precisely what defines the 3-membrane: it is the 4-space which is not empty. It is not continuously full, as it is a cloud of discrete particles like a gas, and the density of particles is very low in most places, but within the 3-membrane it is never zero. The 3-membrane(s) is the surface(s) of 4-polytope(s) with a very large number of vertices (a function of the number of atoms in the universe, if those vertices are enumerated in some manner that includes the universe's plasma matter), and as in any 4-polytope all its elements lie on a 3-dimensional surface (albeit in the case of plasma on a rather inconsequential and rapidly changing 3D surface). === Configurations in the 24-cell === [[File:Reye configuration.svg|thumb|Reye's configuration 12₄16₃ of 12 lines (3 orthogonal groups of 4) intersecting at 16 points.]] The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye’s configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} In its most elemental expression, Reye's configuration is a set of 12 lines which intersect at 16 points, forming two disjoint cubes.{{Efn|The basic expression of Reye's configuration 12₄16₃ of 12 lines and 16 points occurs 12 times in the 24-cell, as the 16 vertices of two opposite (completely orthogonal) cubical cells (in one of the 3 inscribed 8-cells), and the 12 geodesic straight lines (hexagonal great circles) on which the cubes' parallel edges lie (orthogonally in the 3 dimensions of 3-sphere space, embedded in a Euclidean space of 4 orthogonal dimensions).}} It has multiple expressions in the 24-cell.{{Efn|An expression of Reye's configuration 12₄16₃ with respect to isoclinic rotations of the rigid 24-cell: 12 points are reached by 4 half hexagram isoclines in each left ''or'' right 360 degree isoclinic rotation characteristic of the 24-cell, and each of the 16 half hexagram isoclines in the left ''and'' right 360 degree rotations reaches 3 points.}} If the proton and neutron are indeed the hybrid fibrations we have identified, on that assumption we can count the number of valid protons and neutrons that can coexist in the same 24-cell at the same time without colliding, and the total number of distinct configurations of multiple nucleons (the number of distinct nuclides) that a single 24-cell can contain. This will not limit the number of distinct nuclides which can exist, because the 24-cell can be compounded in four-dimensional space in several different ways, but it will quantify the number of nuclides which can occupy the first nuclear shell. We might ask of any less minimal configuration of rotations in a single 24-cell whether it in fact corresponds to an integral number of nucleons occupying the same nuclear shell. A valid ''maximal'' hybrid configuration of rotations in the 24-cell would have the largest number of moving vertices possible without collisions (perhaps 24).{{Efn|The valid maximal configurations do have 24 moving vertices. They include a configuration with 12 vertices rotating ''within'' each of the three 16-cells on 4 octagram isoclines of the same chirality (while remaining within 4 Clifford parallel moving square planes), and visiting all 8 vertices of each 16-cell in each double revolution. The same maximal configuration also has 12 vertices rotating ''among'' the three 16-cells on 4 hexagram isoclines (two left-right pairs) while remaining within 4 Clifford parallel moving hexagonal planes, and visiting all 24 vertices of the 24-cell in each double revolution.}} ====...==== If the particle energies are to be described as the angular momentum of isoclinic rotations of some kind, it is noteworthy that square isoclinic rotations will describe 16-cells, but not (by themselves) the 24-cell. Square isoclinic rotations are associated with the ''internal'' geometry of 16-cells: they are the [[W:16-cell#Rotations|characteristic rotations of the 16-cell]]. The chords of the single [[W:16-cell#Helical construction|isocline of a square isoclinic rotation]] (left or right) are the four orthogonal axes of the 16-cell (enumerated twice), and the isocline is a helical circle passing through all 8 vertices of the 16-cell. A left-right pair of square rotations covers all the elements of the 16-cell (including its 18 great squares and its 16 tetrahedral cells). Thus the square isoclinic rotations say all there is to say about the internal geometry of an individual 16-cell, but they have nothing at all to say about how three 16-cells combine to form a 24-cell. The hexagonal isoclinic rotations, the [[W:24-cell#Isoclinic rotations|characteristic rotations of the 24-cell]], do describe the whole 24-cell. Their hexagram isoclines wind through all three 16-cells, their {{radic|3}} chords connecting the corresponding vertices of pairs of disjoint 16-cells. If vertices moving in hexagonal isoclinic rotations are what carries the energy binding the three quarks together (at least in the case of the neutron), each neutron would require at least two vertices moving in hexagonal isoclinic orbits, for the following reason. We attributed electric charge to the chirality of isospin generally, so we may expect hexagonal isoclinic rotations to contribute to the nucleon's total electric charge, even though they are not intrinsic parts of the three quarks, but rather parts of the whole 24-cell. Since the hexagonal orbits span the three 16-cells equally, the contributions of their moving vertices must be of neutral charge overall. Therefore the neutron must possess pairs of left and right hexagonal isoclinic rotations: minimally, one vertex moving on a left hexagram isocline, and one moving on a right hexagram isocline, which cancel each other because they have exactly opposite isospin. They must be a left and a right isocline from the same fiber bundle, corresponding to the left and right rotations of the same set of Clifford parallel invariant hexagonal planes of rotation. Such a pair is a valid kinematic rotation, because left and right hexagram isoclines of the same fibration do not intersect, and so can never collide.{{Efn|Most, but not all, left and right pairs of isoclinic rotations have isocline pairs which are Clifford parallel and visit disjoint vertex sets. The exception is left and right square isoclinic rotations. Their left and right [[W:16-cell#Helical construction|octagram isoclines]] ''do'' intersect, and they each visit the same set of 8 vertices.}} ====...==== The chiral pair of hexagonal rotations combined with the various square rotations will be a valid hybrid rotation provided no vertex in a hexagram orbit ever collides with a vertex in an octagram orbit. The octagram orbits visit all 8 vertices of the 16-cell in which they are confined. In an up quark with the minimum two moving vertices there are 6 empty vertices at any moment in time, leaving room for cross-traffic. The moving vertex on a hexagram isocline (of which minimally there will be two, a left and a right) will intersect each 16-cell in two places (not antipodal vertices) at different times. If the two moving vertices are antipodal (a moving axis), they will intersect each 16-cell in two axes at different times. There are valid configurations of this set of ''minimal'' hybrid rotations. In up-down-up proton configurations, the 7 moving vertices may be chosen in various ways that avoid collisions. Similarly in down-up-down neutron configurations, there are 6 moving vertices and various valid configurations. These minimal configurations could be are protons and neutrons, or they could be fractional parts of whole nucleons, less minimal configurations of the same kind of hybrid rotation with more moving vertices. These minimal hybrid rotations fall short of the full fibration symmetry of an ordinary isoclinic rotation in various ways. The hexagonally-rotating vertices visit only 12 of the 24 vertices once.{{Efn|Each hexagonal fibration has a right isoclinic rotation on 4 Clifford parallel right hexagram isoclines, and a corresponding left rotation on 4 Clifford parallel left hexagram isoclines, in the same set of 4 Clifford parallel invariant hexagonal planes. The right and left rotations reach disjoint sets of 12 vertices.}} The two square-rotating vertices in each up 16-cell visit all 8 vertices twice; the one in each down 16-cell visits all 8 vertices once. The two hexagram vertices each rotate through three 16-cells, so even with the best synchronization, there will be an oscillation in the total number of moving vertices in each 16-cell at any one time. In these minimal configurations, the 16-cells and the 24-cell would be strangely unbalanced objects. More generally, any such balance would require solutions to the n-body problem for 7 and 6 bodies, respectively. We could attempt to remedy these deficiencies by adding more moving vertices on more isoclines, seeking to make nucleons which are ''hybrid fibrations'' with at least one moving vertex on each isocline of each kind of fibration. This more balanced configuration with complete fibrations can be achieved with 6 vertices moving on hexagram isoclines, synchronized so that there are two of these moving vertices in each 16-cell at once. The proton (or neutron) will be a valid hybrid fibration of the 24-cell if it is synchronized to avoid collisions, with 6 vertices moving over all 24 vertices on isoclines of hexagonal fibrations (3 on left isoclines, and 3 on right isoclines), and 5 (or 4) vertices moving over all 24 vertices on isoclines of square fibrations (4 (or 2) on right isoclines, and 1 (or 2) on left isoclines). The up 16-cells already have two vertices moving on octagram isoclines in the minimal configuration, but the down 16-cells have only one. We can add two vertices moving on a chiral pair of isoclines without breaking the charge balance. Adding two such left-right pairs to each down 16-cell leaves the down 16-cell with two up+right and three down−left vertices. It could decay into an up 16-cell (two up+right vertices) by losing the three down−left vertices with a combined unit negative charge, like the electron emitted during [[W:beta decay|beta minus decay]]. This configuration of a single stable proton (or single unstable neutron) has 9 (or 12) vertices moving on isoclines of square fibrations: 6 (or 6) on right isoclines, and 3 (or 6) on left isoclines. The fact that a proton and neutron form a stable nuclide suggests that they can occupy the same 24-cell together, where their moving vertices combine to stabilize the down 16-cell. This is possible because there is enough room in each 16-cell for it to contain the essential moving vertices of both an up and a down quark at the same time: two vertices moving on an up+right isocline, plus one vertex moving on a down−left isocline provided it is not the complement of either of the right rotations (because the left-right pair of the same distinct rotation have exactly opposite isospin and would cancel each other's charge contribution). This [[W:deuterium|deuterium]] configuration of a stable proton-neutron pair has 9 vertices moving on isoclines of square fibrations: 6 on right isoclines, and 3 on left isoclines. It could be created by [[W:beta plus decay|beta plus decay]] when two protons are forced to occupy the same 24-cell, during the first step in the [[W:proton-proton chain|proton-proton chain]] of [[W:Stellar nucleosynthesis|nucleosynthesis]]. == Wiki researchers == [[W:User:Cloudswrest/Regular_polychoric_rings]], A.P. Goucher (https://cp4space.wordpress.com/2012/09/27/good-fibrations/) [[W:User:Tomruen/Uniform_honeycombs]] [[W:User:PAR]] physicist (stat. mech.) ([[wikipedia:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] in [[wikipedia:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]) [[W:User:DGG]], David Goodman (wikipedia admin, librarian, expert on scientific publishing and open access to published research, first amendment absolutist) [[User:Fedosin|S.Fedosin]] e.g. [[SPФ symmetry]], [[Scale dimension]], '''Model of Gravitational Interaction'''<ref>{{Cite web|url=http://sergf.ru/mgen.htm|title=Model of Gravitational Interaction in the Concept of Gravitons|website=sergf.ru|access-date=2019-05-11}}</ref>, [[Hydrogen system]], [[Metric theory of relativity]] - perhaps begin here<ref>http://vixra.org/pdf/1209.0110v1.pdf</ref> === Fibrations === Currently titled '''Visualization''', perhaps this section could be renamed and multiple references given for the [[W:Hopf fibration]]{{Efn|On a 2-sphere (globe), if you go off in any direction and keep going straight you eventually arrive back at your starting point. Same with the 3-sphere, except you are no longer restricted to a plane, you can go off in any 3-dimensional direction. For the 2-D case all great circles intersect. You can avoid this for the 3-D case. Step away from your initial starting point and go off in a new direction. You want to pick this direction so that you don't intersect the previous geodesic. To this end you have to give your new direction a little "skew" so that your new starting direction is not exactly parallel, and out of the plane, to your old direction. This avoidance of intersection causes the two loops/geodesics to spiral around each other and/or interlock. For the Hopf fibration the farther away you are from the initial starting point, the more "skew" you add. When you are 90 degrees away, you add 90 degrees of skew. This is the most extreme case and you have two interlocking rings passing through the midpoint of the other ring. With this construction you can parameterize the whole 3-sphere, with no two rings ever touching each other. ---- [[W:User:Cloudswrest]] in [[W:Talk:Hopf fibration#Untitled|Talk:Hopf fibration]]|name=|group=}} re: the 24-cell. But beware [[W:User:Cloudswrest/Regular polychoric rings|similar material]] by [[W:User:Cloudswrest]] and [[W:User:Tomruen]] was deleted from the [[W:Hopf fibration]] article once for lack of references. {{Efn|This section was deleted for lacking references. I had been adding some graphics. I copied the section to User:Tomruen/Regular polychoric rings. Tom Ruen (talk) 04:01, 15 November 2014 (UTC) I started the "Discrete Examples" section because of the 120-cell. I consider it a perfect example of the Hopf fibration in a different context. A "physical" example. People might not "get it" when given equations, or theory, or even a continuum picture, but seeing the 120-cell example might provide an "aha moment"! You can SEE it in the Todesco Youtube video of the 120-cell. The other face-to-face cases quickly became obvious. Then somebody mentioned, without any references, the BC helices in the 600-cell, and all the tet based polytope fibrations fell into place also. For the most part math articles have been a pretty safe subject to edit as it's objective, self documenting, except for very esoteric stuff, and people who don't know anything about it are usually uninterested, unlike articles on subjects like say, Martin Luther King, or Hitler, or date rape, where all the social justice warriors and polemicists come out to play. When Eppstein first complained about references over a year ago I did a web search. There are bits and pieces on various web sites and blogs, including some on John Baez's blog, but I could not find any coherent full coverage of the subject, which in any case is pretty obvious to interested parties. But I do know that making snide and sarcastic remarks on the rather competent and prolific work of a Wikipedia math illustrator is over the top. Cloudswrest (talk) 02:00, 17 November 2014 (UTC)}} [https://cp4space.wordpress.com/2012/09/27/good-fibrations/ Goucher] [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] [http://members.home.nl/fg.marcelis/mathemathics.htm Marcelis] The first illustration below shows a torus surface on which 4 equidistant circles lie, each having 4 equidistant points that are 4 of the 16 vertices of a hypercube in stereographic projection. On the vertical line and the circle lie 4 equidistant points each, completing the 16 vertices of the hypercube to the 24 vertices of a 24-cell. === Curiosities === https://physics.info/motion/ https://hexnet.org/ [[w:Double bubble conjecture|Double bubble conjecture]] http://vixra.org/pdf/1812.0482v1.pdf perhaps follows Steinbach's polygonal chord relationships === Review === [https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/ Baez] [http://eusebeia.dyndns.org/4d/bi24dim600cell Who is this?] [http://www.cs.utah.edu/~gk/peek/600slice/index.html Peek software] Who is this? http://eusebeia.dyndns.org If identified perhaps could put under External links on some 4-polytope pages. Especially [http://eusebeia.dyndns.org/4d/vis/vis 4D Visualization] [http://members.home.nl/fg.marcelis/ Marcelis] also [https://fgmarcelis.wordpress.com/ Macelis's other website] === Communities === [http://hi.gher.space/forum/ Higher Space Forum] == 4-space generally == === Dimensional analogy opinion === {{Efn|A [[W:Four-dimensional space#Dimensional analogy|dimensional analogy]] is not a metaphor that we are free to adopt or replace, like the conventional names of the 4-polytopes. The 600-cell is the unique 4-dimensional analogue of the icosahedron in a precise mathematical sense. The symmetry group of the 600-cell is only sometimes called the [[W:Binary icosahedral group|binary ''icosahedral'' group]] (by metaphorical analogy), but the dimensional relationship between the 600-cell and the icosahedron which the operations of the group capture is a mathematical fact (a dimensional analogy). It is not a mistake to call the 600-cell the hexacosichoron or the 4-120-polytope or any other reasonably analogous name we may invent, but it would be a mathematical error to misidentify the 600-cell as the analogue of some other polyhedron than the icosahedron.{{Efn|It is important to distinguish ''dimensional'' analogy from ordinary ''metaphorical'' analogy. ''Dimensional analogy''{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process that can function as a guide to proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction.|name=dimensional analogy}}|name=4-dimensional analogue of the icosahedron|group=}} {{Efn|It is a mistake to confuse the finite mathematics of ''dimensional analogy'' with the infinite art of ''metaphorical analogy''. Dimensional analogy{{Sfn|Coxeter|1973|pp=118-119|loc=§7.1. Dimensional Analogy}} is a rigorous geometric process, like a proof. Problems attacked by this method are frequently intractable when reasoning from ''n'' dimensions to more than ''n'', but it is a [[W:Scientific method|scientific method]] because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction. [https://www.npr.org/books/titles/138359394/what-we-believe-but-cannot-prove-todays-leading-thinkers-on-science-in-the-age-o I believe, but I cannot prove], that there is but one ''correct'' dimensional analogy in every instance; moreover, there is ''always'' that one correct dimensional analogy in every instance (though it may well not have been discovered yet).|name=dimensional analogy}} === Words === [http://os2fan2.com/gloss/index.html The polygloss] Wendy Krieger's glossary of higher-dimensional terms. === Math === The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] {1, ''ξ''₁, ''η'', ''ξ''₂} [Coxeter 1973 p. 216]. Formulas for the conversion of Cartesian coordinates to Hopf coordinates: https://marc-b-reynolds.github.io/quaternions/2017/05/12/HopfCoordConvert.html In 3D every displacement can be reduced to a single-rotation combined with a translation (a screw-displacement). In 4D every displacement can be reduced to either a double rotation or a a single rotation combined with a translation (a 3D screw-displacement).[Coxeter 1973 p 218][[File:Pythagorean theorem - Ani.gif|thumb|caption=(3,4,5) is the smallest [[W:Pythagorean triple|Pythagorean triple]] (a [[W:Special right triangle#Side based|special right triangle]])]] The [[W:Icosian|Icosians]] and [[W:Quaternions]] generally, the "3<small>{{Sfrac|1|2}}</small>-dimensional coordinates of projective 4-space [[https://books.google.com/books?id=5-UlBQAAQBAJ&pg=PA207&lpg=PA207&dq=icosian+ring+golden+field&source=bl&ots=_bR1ndRqQ0&sig=ACfU3U1ePHtwHLEfyG5IC0bQr8Ur736hPw&hl=en&sa=X&ved=2ahUKEwjUiKSPzoriAhWEjp4KHYclD6EQ6AEwCXoECAgQAQ#v=onepage&q=icosian%20ring%20golden%20field&f=false|from google books]] Application of quaternions and projective space generally in the vertex figure space (the curved boundary space of a 4-polytope from the inside). [http://eusebeia.dyndns.org/4d/genrot.pdf Formula for Vector Rotation in Arbitrary Planes] [https://www.facebook.com/Formule.byBNF/posts/httpeusebeiadyndnsorg4dindex/10151772568869171/ Viviani's Theorem] In an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the altitude of the triangle [https://fgmarcelis.wordpress.com/ Macelis's other website] on the geometrical mathematics of physics. === Truncation === What does truncation look like from vertex space (the curved 3-manifold)? Vertices are removed like voids carved out of the interior of the 3-space, or rather, the location of the 3-manifold moves in the 4th direction. How can we imagine observing this (as a continuous process) from the inside of the 3-space? == 120-cell == "Who ordered that?"{{Efn|As the Nobel laureate physicist [[W:Isidor Isaac Rabi|I. I. Rabi]] famously quipped about the unanticipated muon, "Who ordered that?".}} [http://eusebeia.dyndns.org/4d/news2014q1 Omnitruncated 120-cell] - the largest uniform convex 4-polytope. [https://math.okstate.edu/people/segerman/talks/Puzzling_the_120-cell.pdf Segerman] and [https://homepages.warwick.ac.uk/~masgar/Maths/quintessence.pdf Schleimer] A first way to understand the combinatorics of the 120–cell is to look at the layers of dodecahedra at fixed distances from the central dodecahedron. A second way to understand the 120–cell is via a combinatorial version of the [[W:Hopf fibration]]. === Falsified theory === To generate the 120-cell from the 600-cell, it is sufficient to rotate the 600 tetrahedra once, through five positions (either the left-handed or the right-handed chiral rotation). Both chiral rotations are not required, because each 5-click rotation by itself generates all the cells of five ''disjoint'' 600-cells, which together comprise all the vertices of the 120-cell and all ten 600-cells. In other words, the two ways to pick five disjoint 600-cells (out of the ten ''distinct'' 600-cells) correspond to the two sets of opposing tetrahedra in each dodecahedron. FALSIFIED{{Sfn|van Ittersum|2020|loc=§4.3.4 Quaternions with real part 1/2 in each 24-cell in the 600-cell 2I|pp=85-86}} === Dodecahedron coordinates === The red vertices lie at (±φ, ±{{sfrac|1|φ}}, 0) and form a rectangle on the ''xy''-plane. The green vertices lie at (0, ±φ, ±{{sfrac|1|φ}}) and form a rectangle on the ''yz''-plane. The blue vertices lie at (±{{sfrac|1|φ}}, 0, ±φ) and form a rectangle on the ''xz''-plane. (The red, green and blue coordinate triples are circular permutations of each other.) === 30-tetrahedron rings are duals of Petrie polygons === [[W:Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons|Talk:Boerdijk–Coxeter helix#30-tetrahedron rings are duals of Petrie polygons]] == 600-cell == === Rotations === In the 600-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at most 10 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[digon]], and the completely orthogonal fixed plane intersects 0 vertices, 2 vertices (a digon), 4 vertices (a square), or 6 vertices (a hexagon), respectively. === Misc === [[User:Dc.samizdat/600-cell]] The 120 vertices can be seen as the vertices of four sets of 6 orthogonal equatorial pentagons which intersect only at their common center.{{Efn|The edges of the 600-cell form geodesic (great circle) decagons. One can pick out six orthogonal decagons, lying (for example) in the six orthogonal planes of the 4-axis coordinate system. Being completely orthogonal, these decagons share no vertices (they 'miss' each other and intersect at only one point, their common center). Thus they comprise 60 distinct vertices: half the vertices of the 600-cell. By symmetry, the other 60 vertices must occur in an exactly similar (congruent) configuration as another set of six orthogonal decagons (rotated isoclinically with respect to the first set).|name=orthogonal decagons}}{{Efn|Each decagon in the orthogonal set of 6 must share two vertices (a common diameter) with each decagon to which it is not orthogonal (namely, the 66 decagons not in the set). So each set of 6 (orthogonal) decagons populates vertices in 66 (other) decagons. There are 12 sets of 6 orthogonal decagons.}} {{Efn|For any fixed value of 𝜂, we have a 𝜉_''i'' decagon and a 𝜉_''j'' decagon with disjoint vertex sets, because they are completely orthogonal. Conversely, the 6 decagons which intersect at each vertex cannot be mutually orthogonal, and each must have a different value of 𝜂.}} The 600 tetrahedral cells can be seen as the result of a 5-fold subdivision of 24 octahedral cells yielding 120 tetrahedra, in a compound made of 5 such subdivided 24-cells (rotated with respect to each other in angular units of {{sfrac|𝜋|5}}). The 600-cell's edge length is ~0.618 times its radius (the 24-cell's edge length). This is 𝚽, the smaller of the two golden sections of √5. Its reciprocal, the larger golden section, is φ = 1.618. A {{radic|5}} chord will not fit in a polytope of unit radius ({{radic|4}} diameter), but both of its golden sections will fit, and both occur as vertex chords of the unit-radius 600-cell: the smaller 𝚽 as its edge length, and the larger φ as the chord joining vertices that are 3 edge lengths apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 4 sets of) [[wikipedia:24-cell#Hexagons|4 orthogonal hexagons]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 such disjoint sets of 4 orthogonal hexagons will account for all 120 vertices. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 3 sets of) [[wikipedia:24-cell#Squares|6 orthogonal squares]] which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 disjoint sets of 6 such orthogonal squares will account for all 120 vertices. Notice the pentagon inscribed in the decagon. Its {{radic|1.𝚫}} edge chord falls between the {{radic|1}} hexagon and the {{radic|2}} square. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges, evidently dodecahedra), which falls between the 24-cells' envelopes of octahedra (made of {{radic|1}} hexagon edges) and the 8-cells' envelopes of cubes (made of {{radic|2}} square edges). Consider also the {{radic|2.𝚽}} = φ and {{radic|3.𝚽}} chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell.{{Efn|1=The {{radic|2.𝚽}} = <big>φ</big> and {{radic|3.𝚽}} chords produce irregular interior faces and cells, since they make isosceles great circle triangles out of two chords of their own size and one of another size.|name=isosceles chords|group=}} The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell, it also inserts new features deep in the interstices of the [[wikipedia:24-cell#Constructions|24-cell's interior]] structure (which it inherits in full, compounds five-fold, and then elaborates on). evidently the 600-cell has dodecahedra in it golden triangles{{Efn|A [[W:Golden triangle|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ration|golden ratio]] to the distinct side ''b'': : {{sfrac|a|b}} <nowiki>=</nowiki> ϕ <nowiki>=</nowiki> {{sfrac|1 + {{radic|5}}|2}} <nowiki>≈</nowiki> 1.618 It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center.<br> The vertex angle is: : <nowiki>𝛉 = arccos(</nowiki>{{sfrac|ϕ|2}}<nowiki>) = </nowiki>{{sfrac|𝜋|5}}<nowiki> = 36°</nowiki> so the base angles are each {{Sfrac|2𝜋|5}} <nowiki>=</nowiki> 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=golden triangle}} ==== Rotations ==== {{Efn|This is another aspect of the same pentagonal symmetry which permits the partitioning of the 600-cell into [[#Icosahedra|icosahedral clusters]] of 20 cells and clusters of 5 cells.}} Each isoclinic rotation occurs in two chiral forms: there is a Clifford parallel 24-cell to the ''left'' of each 24-cell, and another Clifford parallel 24-cell to its ''right''. The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. ==== Central planes ==== All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart in the 600-cell), hexagon planes ({{sfrac|𝜋|3}} apart in each of 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart in each of 75 inscribed 16-cells). In a 4-polytope, two different central planes may intersect at a common diameter, as they would in 3-space, or they may intersect at a single point only, at the center of the 4-polytope. In the latter case, their great circles are [[W:Clifford parallel|Clifford parallel]].{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space.}} Completely orthogonal{{Efn|name=completely orthogonal planes}} great circles are an example of Clifford parallels, but we can also find non-orthogonal central planes which intersect at only a single point. {{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Because they share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3-dimensions; in fact they are connected like adjacent links in a chain, each passing through the other (without intersecting at any points), forming a [[W:Hopf link|Hopf link]]. === Cayley’s Factorization of 4D Rotations === Any rotation in ℝ⁴ can be seen as the composition of two rotations in a pair of orthogonal two-dimensional subspaces. When the values of the rotation angles in these two subspaces are equal, the rotation is said to be isoclinic. Cayley realized that any rotation in ℝ⁴ can be factored into the commutative composition of two isoclinic rotations.<ref>{{Cite journal|last=Perez-Gracia|first=Alba|last2=Thomas|first2=Federico|date=2016-05-14|title=On Cayley’s Factorization of 4D Rotations and Applications|url=http://dx.doi.org/10.1007/s00006-016-0683-9|journal=Advances in Applied Clifford Algebras|volume=27|issue=1|pages=523–538|doi=10.1007/s00006-016-0683-9|issn=0188-7009}}</ref> The elements of the Lie group of rotations in four-dimensional space, SO(4), can be either simple or double rotations. Simple rotations have a fixed plane (a plane in which all the points are fixed under the rotation), while double rotations have a single fixed point only, the center of rotation. In addition, double rotations present at least a pair of invariant planes that are orthogonal. The double rotation has two angles of rotation, α1 and α2, one for each invariant plane, through which points in the planes rotate. All points not in these planes rotate through angles between α1 and α2. Isoclinic rotations are a particular case of double rotations in which there are infinitely many invariant orthogonal planes, with same rotation angles, that is, α1 = ±α2. These rotations can be left-isoclinic, when the rotation in both planes is the same (α1 = α2), or right-isoclinic, when the rotations in both planes have opposite signs (α1 = −α2). Isoclinic rotation matrices have several important properties: # The composition of two right- (left-) isoclinic rotations is a right- (left-) isoclinic rotation. # The composition of a right- and a left-isoclinic rotation is commutative. # Any 4D rotation can be decomposed into the composition of a right and a left-isoclinic rotation. Hence both form maximal and normal subgroups. Their direct product is a double cover of the group SO(4), as four-dimensional rotations can be seen as the composition of rotations of these two subgroups, and there are two expressions for each element of the group. === 30-gon geodesic === The [[w:600-cell#Geodesics|30-gon vertex-less geodesic of the 600-cell]] reminds me of another remarkable observation about the central axis of the B-C helix made many years ago by the dutch software engineer and geometry experimenter Gerald de Jong,{{Efn|I don't recall de Jong ever writing about 4-dimensional polytopes, but he has a large body of work experimenting with physical and virtual models of geodesics and especially tensegrity structures.}} on a long-extinct email list called ''Synergetics'' that mostly featured discussions of Buckminster Fuller{{Efn|Buckminster Fuller never quite got his mind around 4-polytopes, despite knowing Coxeter, but much of what he observed about the 3-polytopes is directly relevant to the 4-polytopes and original; he had splendid intuition. For example in his obsession with the cuboctahedron (which he called the "vector equilibrium") he was probably the first to sense the real importance of the [[w:24-cell#Radially equilateral honeycomb|radially equilateral]] polytopes. Looking at the 24-cell and tesseract makes me sad for him that he never realized the fourth dimension has a ''regular'' vector equilibrium, one of ''two'' radially equilateral regular polytopes (the other of which is the hypercube!). Just as Fuller's studies and those he inspired (such as [[http://verbchu.blogspot.com/2010/07/ccp-and-hcp-family-of-structures.html%7Cthis]]) are often relevant to the 4-polytopes, the 4-polytopes now inspire new 3-dimensional inventions, such as new forms of Fuller's geodesic domes{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} memes. }} That list didn't extend to 4-polytopes; the geometry discussed there was about 3-dimensional objects, as it also tended to be on [[Magnus Wenninger]]'s ''Polyhedron'' email list. I can't find an archived copy of the email list with Gerald's post but as I recall he studied the B-C helix (Fuller called it the ''tetrahelix'') in 3 dimensions and observed that it had no single central axis, but rather three central axes that passed through each tetrahedron similarly, hitting the volume center of the tetrahedron and hitting two faces near but not at their center, like three holes punched in the face in a small equilateral triangle surrounding the face center. He called the tetrahedra pierced by the three central axes "tetrahedral salt cellars", a wonderfully evocative image and why I have remembered it (correctly, I hope). It is interesting to see that when the helix is bent in the fourth dimension into a ring, in addition to its period being rationalized and its helical edge-paths being straightened into geodesics, its three center axes also merge into one, which passes through a single point at the center of each face. === Golden triangles === The [[W:Golden triangle (mathematics)|golden triangle]] is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions. [[File:Golden_Triangle.svg|right|thumb|A golden triangle. The ratio a:b is equivalent to the golden ratio φ.]] [[w:Golden triangle|Golden triangles]] are found in the nets of several stellations of dodecahedrons and icosahedrons. Since the angles of a triangle sum to 180°, base angles are therefore 72° each.^[1] The golden triangle can also be found in a regular decagon, or an equiangular and equilateral ten-sided polygon, by connecting any two adjacent vertices to the center. This will form a golden triangle. This is because: 180(10-2)/10=144 degrees is the interior angle and bisecting it through the vertex to the center, 144/2=72.^[1] [[File:Kepler_triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the [[w:Golden_ratio|golden ratio]].]] A [[w:Kepler_triangle|Kepler triangle]] is a right triangle with edge lengths in a geometric progression in which the common ratio is √φ, where φ is the golden ratio,^[a] and can be written: , or approximately '''1 : 1.272 : 1.618'''.^[1] The squares of the edges of this triangle are in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between its short side and hypotenuse equal to the golden ratio.^[2] Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed:<blockquote>Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.^[3]</blockquote>Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza,^[4][5] making it a golden pyramid. === Golden chords === The ''golden chords'' demonstrate that <big>ϕ</big> is a circle ratio like <big>𝜋</big>, in fact:<br> : {{sfrac|𝜋|5}} = arccos ({{sfrac|ϕ|2}}) which is one decagon edge. Inversely:<br> : <big>ϕ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})<br> which can be seen from the arc length of the {{radic|2.𝚽}} = <big>ϕ</big> golden chord which is {{sfrac|3𝜋|5}}, but it was apparently discovered first without recourse to geometry.<ref>{{Cite web|title=Pi, Phi and Fibonacci|date=May 15, 2012|author=Gary Meisner |url=https://www.goldennumber.net/pi-phi-fibonacci/|postscript=: Robert Everest discovered that you can express <big>ϕ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series: <big>ϕ</big> = 1 – 2 cos (3𝜋/5)}}</ref> Phi is a circle ratio, like Pi Pi = 5 arccos (.5 Phi) Note: The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians. Note: Above formulas expressed in radians, not degrees Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows: 5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series: Phi = 1 – 2 cos ( 3 Pi / 5) Golden ratio of chords: Peter Steinbach * Golden Fields - DPF formula https://www.jstor.org or https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1997.11996494 * Sections Beyond Golden https://archive.bridgesmathart.org/2000/bridges2000-35.pdf Sacred cut (of octagon) https://archive.bridgesmathart.org/2011/bridges2011-559.pdf [[W:George Phillips Odom Jr.]] discovery of golden section in the mid-edge-bisectors of the tetrahedron (applies to the construction of the 600-cell from the 24-cell via truncation of its central tetrahedra) - also look at the other circle/triangle and sphere/tetrahedra relationships he discovered - footnote his relationship to Coxeter and Conway [[w:Intersecting chords theorem|Intersecting chords theorem]] === Nonconvex regular decagon === [[File:Golden_tiling_with_rotational_symmetry.svg|left|thumb|This '''[[Tessellation|tiling]]''' by '''[[Golden triangle (mathematics)|golden]]''' triangles, a regular '''[[pentagon]]''', contains a '''[[wikipedia:Stellation|stellation]]''' of '''[[Regular polygon|regular]] decagon''', the '''[[Schläfli symbol|Schäfli&nbsp;symbol]]''' of which is {10/3}.]] The length [[ratio]] of two inequal edges of a golden triangle is the [[golden ratio]], denoted <math>\text{by }\Phi \text{,}</math> or its [[Multiplicative inverse|multiplicative&nbsp;inverse]]: :<math> \Phi - 1 = \frac{1}{\Phi} = 2\,\cos 72\,^\circ = \frac{1}{\,2\,\cos 36\,^\circ} = \frac{\,\sqrt{5} - 1\,}{2} \text{.}</math> So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this [[Star polygon|star&nbsp;polygon]].{{Clear}} == 24-cell == Visualize the three 16-cells inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} [[File:24-cell graph D4.svg|thumb]] Whoever this is has some beautiful illustrations of closely related polytopes e.g.<ref>http://eusebeia.dyndns.org/4d/rect24cell</ref> [http://members.home.nl/fg.marcelis/24-cell.htm Marcelis] including [http://members.home.nl/fg.marcelis/24celiso.htm#gem the gem of the modular universe] === Cuboctahedron === In the 24-cell, there are 16 great hexagons, and each has one pair of vertices (one diameter) which lies on a Cartesian coordinate system axis (in the vertex-up frame of reference). The 24-cell can be seen as the 24 vertices of four orthogonal great hexagons, each aligned with one of the four orthogonal coordinate system axes, and each contributing 6 disjoint vertices. In four dimensions, the four ''orthogonal'' planes do not intersect except at their common center. Each great hexagon does not share any vertices with the 3 other hexagons to which it is orthogonal,{{Efn|In four dimensions up to 6 planes through a common point may be mutually orthogonal. The 18 great squares of the 24-cell comprise three sets of 6 orthogonal planes. The 16 great hexagons, however, comprise four sets of just 4 orthogonal planes. In four dimensions there is both a symmetrical arrangement of 6 orthogonal planes, and a symmetrical arrangement of 4 orthogonal planes. We can pick out 6 orthogonal squares in the 16-cell, 8-cell, or 24-cell, but the symmetry of 4 orthogonal hexagons emerges only in the 24-cell.}} but it shares two vertices with each of the 12 other great hexagons to which it is ''not'' orthogonal. Four hexagonal geodesics pass though each vertex. In the cuboctahedron, there are four great hexagons, but they are not orthogonal. Each intersects with each of the others in two vertices (one diameter), and two hexagons pass through each vertex. (In three dimensions, two planes through a common point intersect in a line, whether they are orthogonal or not.) At most one of the four hexagons can have a pair of vertices (a diameter) which lies on a coordinate system axis. In such a frame of reference, the other two axes pass through the centers of a pair of opposing triangular faces, and through the centers of a pair of opposing edges, respectively. === Geometry === ==== Triangles ==== ...to be added:<br> If the dual of the [[W:24-cell#Squares|24-cell of edge length {{radic|2}}]] is taken by reciprocating it about its ''circumscribed'' sphere, another 24-cell is found which has edge length and circumradius {{radic|3}}, and its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows: :<math>(0, \pm 1, \pm 1, \pm 1) \in \mathbb{R}^4</math> The 4 orthogonal planes in which the 8 triangles lie are ''not'' orthogonal planes of this coordinate system. The triangles' {{radic|3}} edges are the ''diagonals'' of cubical ''cells'' of this coordinate lattice.{{Efn|For example: {{green|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> are the two opposing central triangles on the ''y'' axis (in this coordinate system of edge length {{radic|3}}).|name=|group=}} The 24 vertices are also the vertices of 96 ''other'' triangles of edge length {{radic|3}} that occur in 48 parallel pairs, in planes one edge length apart. Each plane sections the 24-cell through three vertices but does not pass through the center.{{Efn|...add coordinates example to existing note...|name=|group=}} ..see [[w:5-cell#Construction|5-cell#Construction]]: Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2{{radic|2}}: :<math>\left( 1,1,1, \frac{-1}\sqrt{5} \right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( -1,-1,1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0,0,0,\sqrt{5}-\frac 1\sqrt{5} \right)</math> ... obs: <br> Add coordinate examples to this footnote{{Efn|Each of these 96 triangular planes sections the 24-cell <small>{{sfrac|1|2}}</small> edge-length below a vertex, and <small>{{sfrac|1|2}}</small> edge-length above the center, measured from the center of the triangle, which is on a 24-cell diameter joining two opposite vertices. However, these paired parallel triangular planes are not orthogonal to the diameter line; they are inclined with respect to it. Each plane contains only one triangle (unlike the central hexagonal planes with their two opposing triangles), but they occur in co-centric sets of four, inclined different ways about the diameter line. The 96 triangles are inclined both with respect to the coordinate system's 6 orthogonal planes (the 6 perpendicular squares) and with respect to the hexagons. Each triangle contains one vertex from a square, and two from different hexagons. Thus their Cartesian coordinates take many different forms, but as examples: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> is one of the four face triangles with one vertex at the positive vertex on the ''y'' axis; and below is the opposite face of one of the two tetrahedra it is a face of, inclined about the negative part of the ''y'' axis: {{color|cyan|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)}} {{color|orange|{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{green|{{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)}} {{color|orange|{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)}}<br> |name=non-central triangles}} === Rotations === The vertices of a convex 4-polytope lie on the [[W:3-sphere|3-sphere]], so alternatively they can be described using 4-dimensional spherical coordinates such as the [[w:N-sphere#Hopf_coordinates|Hopf coordinates]] {''r'', ''ξ''₁, ''η'', ''ξ''₂}, which reveal more structure. For the 24-cell of edge length and radius 1, the Hopf coordinates of its vertices can be obtained by permuting the three angle coordinates as follows: {1, ±{{sfrac|𝜋|3}}, ±{{sfrac|𝜋|2}}, ±{{sfrac|2𝜋|3}}}[[File:30-60-90.svg|thumb]] in 3D section of a 24-cell (of edge length 2) we can construct a tetrahedron with 2{{radic|3}} legs descending from a vertex (0,0,2) with its base plane triangle intersecting the 2-sphere at (x,y,-1), but only if we put the base vertices at distances apart less than 2{{radic|3}}, as a 2{{radic|2}} 3 3 isosceles triangle e.g. with these vertices: (-{{radic|2}},-1,-1) <--- 2{{radic|2}} --> ((+{{radic|2}},-1,-1)) <-- 3 --> (0, (+{{radic|3}}, -1) <-- 3 --> Apparently the base vertices of the tetrahedron are displaced out of this hyperplane in the 4th dimension so the base edges are foreshortened. ... and eight meet at the volume center of each tesseract cube{{Efn|The geometry of the tesseract cube volume centers is exactly the same as the vertices: a cubical vertex figure, in which four long diameters cross at the center. They are 24 interior vertices, arrayed as a 1/2 size 24-cell around the central interior vertex, at the midpoints of the 24 unit-length radii (which pass through opposite face centers of the vertex figure).}} {{Clear}} ==Kepler problem== [[File:Kepler-solar-system-2.png|thumb|Detailed view of the inner sphere of Kepler's Mysterium Cosmographicum.]]We are apt to be smug about the quaint mythological phantasies of our great forebears, as when we learn that Issac Newton worked for more than 30 years as an alchemist trying to turn base metals into gold, before he was appointed Chancellor of the Exchequer to preside over England's mint of sterling instead. [[W:Mysterium Cosmographicum|Kepler's astrolabe]] of Plato's holy solids looks that way to us, like a religious miracle the great astronomer hoped for, before he discovered the real symmetry in his three great conservation laws of motion. It should humble us, then, to find out that he was on to something deeper all along, and make us wonder all the more at his genius, that it was the SO(4) rotational symmetry he glimpsed, which generates those conservation laws by Noether's theorem, and also generates the 4-polytopes that Schläfli would discover on the 3-sphere in 4-space two and a half centuries after Kepler, who somehow imagined them from below, projected on the 2-sphere in 3-space. ==Laplace–Runge–Lenz vector== the following rescued from [[W:Laplace–Runge–Lenz vector]] version of 23:32, 27 November 2006 from which it was removed by WillowW (talk | contribs) at 23:46, 28 November 2006 (the Moebius transform was fun, but needs to go now) ===Intuitive picture of the rotations in four dimensions=== [[Image:Kepler_hodograph_family_transformed.png|thumb|right|280px|Figure 4: [[W:Circle inversion|Inversion]] in the dotted black circle of Figure 3 transforms the family of circular hodographs of a given energy ''E'' into a family of straight lines intersecting at the same point. Thus, the orbits of the same energy but different angular momentum can be transformed into one another by a simple rotation.]] The simplest way to visualize the particular symmetry of the Kepler problem is through its [[W:hodograph|hodograph]]s, the perfectly circular traces of the momentum vector (Figures 2 and 3). For a given total [[W:energy|energy]] ''E'', the hodographs are circles centered on the ''p_y''-axis, all of which intersect the ''p_x''-axis at the same two points, ''p_x=±p₀'' (Figure 3). To eliminate the normal rotational symmetry, the coordinate system has been fixed so that the orbit lies in the ''x''-''y'' plane, with the major semiaxis aligned with the ''x''-axis. [[W:Circle inversion|Inversion]] centered on one of the foci transforms the hodographs into straight lines emanating from the inversion center (Figure 4). These straight lines can be converted into one another by a simple two-dimensional rotation about the inversion center. Thus, all orbits of the same energy can be continuously transformed into one another by a rotation that is independent of the normal three-dimensional rotations of the system; this represents the "higher" symmetry of the Kepler problem. == Formatting idioms == :{{sfrac|1|2}}(±φ𝞍ϕ𝜙𝝓𝚽𝛷𝜱, ±1, ±{{sfrac|1|φ}}, 0). : :360^{\circ} :denoted <math>\tbinom{24}{4}</math>   == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book)}} * {{Cite book | last1=Coxeter | first1=H.S.M. | author1-link=W:Harold Scott MacDonald Coxeter | last2=du Val | first2=Patrick | author2-link=W:Patrick du Val | last3=Flather | first3=H.T. | last4=Petrie | author4-link=W:John Flinders Petrie | first4=J.F. | year=1938 | title=[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]] | publisher=University of Toronto Studies (Mathematical Series) | volume=6 }} * {{Cite book|title=The Coxeter Legacy|chapter=Coxeter Theory: The Cognitive Aspects|last=Borovik|first=Alexandre|year=2006|publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|pp=17-43|ISBN=978-0821837221|url=https://www.academia.edu/26091464/Coxeter_Theory_The_Cognitive_Aspects}} * {{Cite journal | last=Miyazaki | first=Koji | year=1990 | title=Primary Hypergeodesic Polytopes | journal=International Journal of Space Structures | volume=5 | issue=3–4 | pages=309–323 | doi=10.1177/026635119000500312 | s2cid=113846838 }} * {{Cite book|url=https://link.springer.com/chapter/10.1007/978-981-10-7617-6_6|title=Nanoinformatics|last=Nishio|first=Kengo|last2=Miyazaki|first2=Takehide|publisher=Springer|year=2018|isbn=|editor-last=Tanaka|location=Singapore|pages=97-130|chapter=Polyhedron and Polychoron Codes for Describing Atomic Arrangements}} * {{Cite journal|last=Waegell|first=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|url=https://arxiv.org/abs/0911.2289v2|language=en|doi=10.1088/1751-8113/43/10/105304}} * {{Cite journal|last=Sadoc|first=Jean-Francois|date=2001|title=Helices and helix packings derived from the {3,3,5} polytope|journal=[[W:European Physical Journal E|European Physical Journal E]]|volume=5|pages=575–582|doi=10.1007/s101890170040|doi-access=free|s2cid=121229939|url=https://www.researchgate.net/publication/260046074}} {{Refend}} <references group="lower-alpha" /> clq6ebyzh7f96qu3lb2ir40wkcl172g 0 249667 2691696 2679934 2024-12-12T22:19:00Z Dc.samizdat 2856930 Categories 2691696 wikitext text/x-wiki {{Short description|Four-dimensional analog of the dodecahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=120-cell | Image_File=Schlegel wireframe 120-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Snub 24-cell|31]] | Index=32 | Next=[[W:Rectified 120-cell|33]] | Schläfli={5,3,3}| CD={{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}| Cell_List=120 [[W:Dodecahedron|{5,3}]] [[Image:Dodecahedron.png|20px]]| Face_List=720 [[W:Pentagon|{5}]] [[File:Regular pentagon.svg|20px]]| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=[[W:Triacontagon|30-gon]]| Coxeter_Group=H₄, [3,3,5]| Vertex_Figure=[[File:120-cell verf.svg|80px]]<br>[[W:Tetrahedron|tetrahedron]]| Dual=[[600-cell]]| Property_List=[[W:Convex set|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:120-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:Geometry|geometry]], the '''120-cell''' is the [[W:Convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {5,3,3}. It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone|Pileio|Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone|Pileio|Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains a compound of several of these great circle polygons in the same central plane, illustrated separately.{{Efn|name=irregular great dodecagon}} An implication of the compounding is that the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an irregular great dodecagon which also has #4 chord edges.{{Efn|name=irregular great dodecagon}} In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in the same rotation planes, the hexagonal central planes of the 24-cell.{{Efn|name=edge rotation planes}}]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|0.𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an irregular great dodecagon, a compound [[#Chords|great circle polygon of the 120-cell]] which is illustrated separately.{{Efn|name=irregular great dodecagon}}|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 #4 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel {12} great circle polygons.{{Efn|name=irregular great dodecagon}}]]The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord connects two vertices which are ''n'' edge lengths apart on a Petrie polygon. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we name the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ or decagrams [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which are Petrie polygons that circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two adjacent great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 | |colspan=2|120-cell edge{{Efn|name=120-cell Petrie {30}-gon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|0.𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|{{radic|2}} φ²}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|0.𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} |{{Efn|name=irregular great dodecagon}} |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different central planes.{{Efn|name=same 200 planes}}|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400{{Efn|name=same 200 planes}} |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>16200 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer|Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with distinct shapes from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}.{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an irregular dodecagon,{{Efn|name=irregular great dodecagon}} {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and {{Background color|gainsboro|<big>☐</big> planes that intersect {4} vertices}} in several kinds of rectangle, including a square. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[W:#Concentric hulls|polyhedral sections]] of the 120-cell, beginning with a vertex, the 0₀ section. The correspondence is that each 120-cell vertex is surrounded by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius.{{Efn|In the curved 3-dimensional space of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} The #1 chord is the "radius" of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the "radius" of its congruent opposing 29₀ section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 15₀ sections are coincident. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=11|30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- !colspan=4|Short chord !colspan=2|Great circle polygons !Rotation !colspan=4|Long chord |- style="background: palegreen;"| |rowspan=2|1₀<br><br>#1 |{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |colspan=2|<math>1 / \phi^2\sqrt{2}</math> |rowspan=2|[[File:Irregular great hexagons of the 120-cell.png|100px]] |rowspan=2|400 irregular great hexagons{{Efn|name=irregular great dodecagon}} / 4<br> (600 great rectangles)<br> in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |colspan=2|<math>\phi^{5}\sqrt{3} / \sqrt{8}</math> |rowspan=2|29₀<br><br>#14 |- style="background: palegreen;"| |15.5~° |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |0.270~ |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: gainsboro;"| |rowspan=2|2₀<br><br>#2 |{{Efn|name=#2 chord}} |colspan=2|<math>1 / \phi\sqrt{2}</math> |rowspan=2|[[File:25.2° × 154.8° chords great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2|4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |colspan=2| |rowspan=2|28₀<br><br>#13 |- style="background: gainsboro;"| |25.2~° |{{radic|0.19~}} |0.437~ |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: yellow;"| |rowspan=2|3₀<br><br>#3 |<math>\pi / 5</math> |colspan=2|<math>1 / \phi</math> |rowspan=2|[[File:Great decagon rectangle.png|100px]] |rowspan=2|720 great decagons / 12<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes |rowspan=2|5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |colspan=2|<math>\sqrt{2+\phi}</math> |rowspan=2|27₀<br><br>#12 |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: gainsboro;"| |rowspan=2|4₀<br><br>#4−1 | |colspan=2|<math>\sqrt{1}/\sqrt{2}</math> |rowspan=2|[[File:√0.5 × √3.5 great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2|<math>\sqrt{7} / \sqrt{2}</math> |rowspan=2|26₀<br><br>#11+1 |- style="background: gainsboro;"| |41.4~° |{{radic|0.5}} |0.707~ |138.6~° |{{radic|3.5}} |1.871~ |- style="background: palegreen;"| |rowspan=2|5₀<br><br>#4 | |colspan=2|<math>\sqrt{3} / \phi\sqrt{2}</math> |rowspan=2|[[File:Irregular great dodecagon.png|100px]] |rowspan=2|200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown).{{Efn|name=irregular great dodecagon}} Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}} / 4<br>(600 great rectangles)<br>in 200 △ planes |rowspan=2|{{Efn|name=#4 isocline chord bridge}} | |colspan=2|<math>\phi^2 / \sqrt{2}</math> |rowspan=2|25₀<br><br>#11 |- style="background: palegreen;"| |44.5~° |{{radic|0.57~}} |0.757~ |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: gainsboro; height:50px"| |rowspan=2|6₀<br><br>#4+1 | |colspan=2| |rowspan=2|[[File:49.1° × 130.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|24₀<br><br>#11−1 |- style="background: gainsboro;"| |49.1~° |{{radic|0.69~}} |0.831~ |130.9~° |{{radic|3.31~}} |1.819~ |- style="background: gainsboro; height:50px"| |rowspan=2|7₀<br><br>#5−1 | |colspan=2| |rowspan=2|[[File:56° × 124° great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|23₀<br><br>#10+1 |- style="background: gainsboro;"| |56° |{{radic|0.88~}} |0.939~ |124° |{{radic|3.12~}} |1.766~ |- style="background: palegreen;"| |rowspan=2|8₀<br><br>#5 |<math>\pi / 3</math> |colspan=2| |rowspan=2|[[File:Great hexagon.png|100px]] |rowspan=2|400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}} / 16<br> (1200 great rectangles)<br>in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<math>2\pi / 3</math> |colspan=2| |rowspan=2|22₀<br><br>#10 |- style="background: palegreen;"| |60° |{{radic|1}} |1 |120° |{{radic|3}} |1.732~ |- style="background: gainsboro; height:50px"| |rowspan=2|9₀<br><br>#5+1 | |colspan=2| |rowspan=2|[[File:66.1° × 113.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|21₀<br><br>#10−1 |- style="background: gainsboro;"| |66.1~° |{{radic|1.19~}} |1.091~ |113.9~° |{{radic|2.81~}} |1.676~ |- style="background: gainsboro; height:50px"| |rowspan=2|10₀<br><br>#6−1 | |colspan=2| |rowspan=2|[[File:69.8° × 110.2° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|20₀<br><br>#9+1 |- style="background: gainsboro;"| |69.8~° |{{radic|1.31~}} |1.144~ |110.2~° |{{radic|2.69~}} |1.640~ |- style="background: yellow;"| |rowspan=2|11₀<br><br>#6 |<math>2\pi/5</math> |colspan=2|<math>\sqrt{3-\phi}</math> |rowspan=2|[[File:Great pentagons rectangle.png|100px]] |rowspan=2|1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}} / 12<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes |rowspan=2|4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |colspan=2|<math>\phi</math> |rowspan=2|19₀<br><br>#9 |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |108° |{{radic|2.𝚽}} |1.618~ |- style="background: palegreen; height:50px"| |rowspan=2|12₀<br><br>#6+1 | |colspan=2|<math>\sqrt{3} / \sqrt{2}</math> |rowspan=2|[[File:Great 5-cell digons rectangle.png|100px]] |rowspan=2|1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}} / 4<br>(600 great rectangles)<br> in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |colspan=2|<math>\sqrt{5} / \sqrt{2}</math> |rowspan=2|18₀<br><br>#8 |- style="background: palegreen;"| |75.5~° |{{radic|1.5}} |1.224~ |104.5~° |{{radic|2.5}} |1.581~ |- style="background: gainsboro; height:50px"| |rowspan=2|13₀<br><br>#6+2 | |colspan=2| |rowspan=2|[[File:81.1° × 98.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|17₀<br><br>#8−1 |- style="background: gainsboro;"| |81.1~° |{{radic|1.69~}} |1.300~ |98.9~° |{{radic|2.31~}} |1.520~ |- style="background: gainsboro; height:50px"| |rowspan=2|14₀<br><br>#7−1 | |colspan=2| |rowspan=2|[[File:84.5° × 95.5° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|16₀<br><br>#7+1 |- style="background: gainsboro;"| |84.5~° |{{radic|0.81~}} |1.345~ |95.5~° |{{radic|2.19~}} |1.480~ |- style="background: gainsboro;"| |rowspan=2|15₀<br><br>#7 |<math>\pi / 2</math> |colspan=2| |rowspan=2|[[File:Great square rectangle.png|100px]] |rowspan=2|4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}} / 27<br> in 4050 <big>☐</big> planes |rowspan=2|4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |colspan=2| |rowspan=2|15₀<br><br>#7 |- style="background: gainsboro;"| |90° |{{radic|2}} |1.414~ |90° |{{radic|2}} |1.414~ |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons / 4".}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1, 2, & 7 are each overlapping pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hulls 6 & 8 are pairs of [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.]] {{Clear}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the length of the edge (the '''#1 [[#Chords|chord]]''' of the 120-cell) is {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270. Eight {{Color|orange}} vertices lie at the Cartesian coordinates (±φ³{{radic|8}}, ±φ³{{radic|8}}, ±φ³{{radic|8}}) relative to origin at the cell center. They form a cube (dashed lines) of edge length {{sfrac|1|φ{{radic|2}}}} ≈ 0.437 (the pentagon diagonal, and the '''#2 chord''' of the 120-cell). The face diagonals of the cube (not shown) of edge length {{sfrac|1|φ}} ≈ 0.618 are the edges of tetrahedral cells inscribed in the cube (600-cell edges, and the '''#3 chord''' of the 120-cell). The diameter of the dodecahedron is {{sfrac|{{radic|3}}|φ{{radic|2}}}} ≈ 0.757 (the cube diagonal, and the '''#4 chord''' of the 120-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula]|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer|Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer|Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an irregular dodecagon in a central plane.{{Efn|[[File:Great dodecagon of the 120-cell.png|thumb|200px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]]The 120-cell has an irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] of 6 edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} The irregular great dodecagon contains two irregular great hexagons ({{color|red|red}}) inscribed in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The twelve regular hexagon edges (#5 chords), the six cell-diameter edges of the dodecagon (#4 chords), and the six 120-cell edges of the dodecagon (#1 chords), are all chords of the same great circle, but the other 24 zig-zag edges (#1 chords, not shown) that bridge the six #4 edges of the dodecagon do not lie in this great circle plane. The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of dodecagon planes. The 120-cell contains 200 such {12} central planes (100 completely orthogonal pairs), the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in each of the 10 inscribed 600-cells.{{Efn|The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} All ten 600-cells occupy the same set of 200 irregular great dodecagon central planes.{{Efn|name=irregular great dodecagon}} There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint great hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells.|name=same 200 planes}}|name=irregular great dodecagon}} Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each irregular great dodecagon, in alternate positions.{{Efn|name=irregular great dodecagon}} === Perspective projections === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} As in all the illustrations in this article, only the edges of the 120-cell appear in these renderings. All the other [[#Chords|chords]] are not shown. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them. These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (below left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating animations above, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Orthogonal projections === [[W:Orthogonal projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} 3-dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image. {| class="wikitable" style="width:540px;" |+3D orthographic projections |[[File:120Cell 3D.png|270px]]<br>3D isometric projection |align=center|[[File:Cell120.ogv|270px]]<br>Animated 4D rotation |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell === The '''Davis 120-cell''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist}} ==Citations== {{Reflist}} ==References== {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book)}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd | isbn= }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | place= | edition=2nd | isbn=978-0-471-01003-6 | url=https://www.wiley.com/en-us/Kaleidoscopes%3A+Selected+Writings+of+H+S+M+Coxeter-p-9780471010036 | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic }} ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] r6oae7yyzeo5z0vk7cuw42kof2p0oz0 2691762 2691696 2024-12-13T08:21:24Z Dc.samizdat 2856930 2691762 wikitext text/x-wiki {{Short description|Four-dimensional analog of the dodecahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=120-cell | Image_File=Schlegel wireframe 120-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Snub 24-cell|31]] | Index=32 | Next=[[W:Rectified 120-cell|33]] | Schläfli={5,3,3}| CD={{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}| Cell_List=120 [[W:Dodecahedron|{5,3}]] [[Image:Dodecahedron.png|20px]]| Face_List=720 [[W:Pentagon|{5}]] [[File:Regular pentagon.svg|20px]]| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=[[W:Triacontagon|30-gon]]| Coxeter_Group=H₄, [3,3,5]| Vertex_Figure=[[File:120-cell verf.svg|80px]]<br>[[W:Tetrahedron|tetrahedron]]| Dual=[[600-cell]]| Property_List=[[W:Convex set|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:120-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:Geometry|geometry]], the '''120-cell''' is the [[W:Convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {5,3,3}. It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone|Pileio|Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone|Pileio|Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains a compound of several of these great circle polygons in the same central plane, illustrated separately.{{Efn|name=irregular great dodecagon}} An implication of the compounding is that the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an irregular great dodecagon which also has #4 chord edges.{{Efn|name=irregular great dodecagon}} In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in the same rotation planes, the hexagonal central planes of the 24-cell.{{Efn|name=edge rotation planes}}]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|0.𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an irregular great dodecagon, a compound [[#Chords|great circle polygon of the 120-cell]] which is illustrated separately.{{Efn|name=irregular great dodecagon}}|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 #4 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel {12} great circle polygons.{{Efn|name=irregular great dodecagon}}]]The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord connects two vertices which are ''n'' edge lengths apart on a Petrie polygon. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we name the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ or decagrams [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which are Petrie polygons that circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two adjacent great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 | |colspan=2|120-cell edge{{Efn|name=120-cell Petrie {30}-gon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|0.𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|{{radic|2}} φ²}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|0.𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} |{{Efn|name=irregular great dodecagon}} |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different central planes.{{Efn|name=same 200 planes}}|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400{{Efn|name=same 200 planes}} |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>16200 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer|Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with distinct shapes from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}.{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an irregular dodecagon,{{Efn|name=irregular great dodecagon}} {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and {{Background color|gainsboro|<big>☐</big> planes that intersect {4} vertices}} in several kinds of rectangle, including a square. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[W:#Concentric hulls|polyhedral sections]] of the 120-cell, beginning with a vertex, the 0₀ section. The correspondence is that each 120-cell vertex is surrounded by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius.{{Efn|In the curved 3-dimensional space of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} The #1 chord is the "radius" of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the "radius" of its congruent opposing 29₀ section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 15₀ sections are coincident. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=11|30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- !colspan=4|Short chord !colspan=2|Great circle polygons !Rotation !colspan=4|Long chord |- style="background: palegreen;"| |rowspan=2|1₀<br><br>#1 |{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |colspan=2|<math>1 / \phi^2\sqrt{2}</math> |rowspan=2|[[File:Irregular great hexagons of the 120-cell.png|100px]] |rowspan=2|400 irregular great hexagons{{Efn|name=irregular great dodecagon}} / 4<br> (600 great rectangles)<br> in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |colspan=2|<math>\phi^{5}\sqrt{3} / \sqrt{8}</math> |rowspan=2|29₀<br><br>#14 |- style="background: palegreen;"| |15.5~° |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |0.270~ |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: gainsboro;"| |rowspan=2|2₀<br><br>#2 |{{Efn|name=#2 chord}} |colspan=2|<math>1 / \phi\sqrt{2}</math> |rowspan=2|[[File:25.2° × 154.8° chords great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2|4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |colspan=2| |rowspan=2|28₀<br><br>#13 |- style="background: gainsboro;"| |25.2~° |{{radic|0.19~}} |0.437~ |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: yellow;"| |rowspan=2|3₀<br><br>#3 |<math>\pi / 5</math> |colspan=2|<math>1 / \phi</math> |rowspan=2|[[File:Great decagon rectangle.png|100px]] |rowspan=2|720 great decagons / 12<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes |rowspan=2|5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |colspan=2|<math>\sqrt{2+\phi}</math> |rowspan=2|27₀<br><br>#12 |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: gainsboro;"| |rowspan=2|4₀<br><br>#4−1 | |colspan=2|<math>\sqrt{1}/\sqrt{2}</math> |rowspan=2|[[File:√0.5 × √3.5 great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2|<math>\sqrt{7} / \sqrt{2}</math> |rowspan=2|26₀<br><br>#11+1 |- style="background: gainsboro;"| |41.4~° |{{radic|0.5}} |0.707~ |138.6~° |{{radic|3.5}} |1.871~ |- style="background: palegreen;"| |rowspan=2|5₀<br><br>#4 | |colspan=2|<math>\sqrt{3} / \phi\sqrt{2}</math> |rowspan=2|[[File:Irregular great dodecagon.png|100px]] |rowspan=2|200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown).{{Efn|name=irregular great dodecagon}} Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}} / 4<br>(600 great rectangles)<br>in 200 △ planes |rowspan=2|{{Efn|name=#4 isocline chord bridge}} | |colspan=2|<math>\phi^2 / \sqrt{2}</math> |rowspan=2|25₀<br><br>#11 |- style="background: palegreen;"| |44.5~° |{{radic|0.57~}} |0.757~ |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: gainsboro; height:50px"| |rowspan=2|6₀<br><br>#4+1 | |colspan=2| |rowspan=2|[[File:49.1° × 130.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|24₀<br><br>#11−1 |- style="background: gainsboro;"| |49.1~° |{{radic|0.69~}} |0.831~ |130.9~° |{{radic|3.31~}} |1.819~ |- style="background: gainsboro; height:50px"| |rowspan=2|7₀<br><br>#5−1 | |colspan=2| |rowspan=2|[[File:56° × 124° great rectangle.png|100px]] |rowspan=2|Great rectangles<br>in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|23₀<br><br>#10+1 |- style="background: gainsboro;"| |56° |{{radic|0.88~}} |0.939~ |124° |{{radic|3.12~}} |1.766~ |- style="background: palegreen;"| |rowspan=2|8₀<br><br>#5 |<math>\pi / 3</math> |colspan=2| |rowspan=2|[[File:Great hexagon.png|100px]] |rowspan=2|400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}} / 16<br> (1200 great rectangles)<br>in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<math>2\pi / 3</math> |colspan=2| |rowspan=2|22₀<br><br>#10 |- style="background: palegreen;"| |60° |{{radic|1}} |1 |120° |{{radic|3}} |1.732~ |- style="background: gainsboro; height:50px"| |rowspan=2|9₀<br><br>#5+1 | |colspan=2| |rowspan=2|[[File:66.1° × 113.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|21₀<br><br>#10−1 |- style="background: gainsboro;"| |66.1~° |{{radic|1.19~}} |1.091~ |113.9~° |{{radic|2.81~}} |1.676~ |- style="background: gainsboro; height:50px"| |rowspan=2|10₀<br><br>#6−1 | |colspan=2| |rowspan=2|[[File:69.8° × 110.2° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|20₀<br><br>#9+1 |- style="background: gainsboro;"| |69.8~° |{{radic|1.31~}} |1.144~ |110.2~° |{{radic|2.69~}} |1.640~ |- style="background: yellow;"| |rowspan=2|11₀<br><br>#6 |<math>2\pi/5</math> |colspan=2|<math>\sqrt{3-\phi}</math> |rowspan=2|[[File:Great pentagons rectangle.png|100px]] |rowspan=2|1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}} / 12<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes |rowspan=2|4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |colspan=2|<math>\phi</math> |rowspan=2|19₀<br><br>#9 |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |108° |{{radic|2.𝚽}} |1.618~ |- style="background: palegreen; height:50px"| |rowspan=2|12₀<br><br>#6+1 | |colspan=2|<math>\sqrt{3} / \sqrt{2}</math> |rowspan=2|[[File:Great 5-cell digons rectangle.png|100px]] |rowspan=2|1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}} / 4<br>(600 great rectangles)<br> in 200 △ planes |rowspan=2|4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |colspan=2|<math>\sqrt{5} / \sqrt{2}</math> |rowspan=2|18₀<br><br>#8 |- style="background: palegreen;"| |75.5~° |{{radic|1.5}} |1.224~ |104.5~° |{{radic|2.5}} |1.581~ |- style="background: gainsboro; height:50px"| |rowspan=2|13₀<br><br>#6+2 | |colspan=2| |rowspan=2|[[File:81.1° × 98.9° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|17₀<br><br>#8−1 |- style="background: gainsboro;"| |81.1~° |{{radic|1.69~}} |1.300~ |98.9~° |{{radic|2.31~}} |1.520~ |- style="background: gainsboro; height:50px"| |rowspan=2|14₀<br><br>#7−1 | |colspan=2| |rowspan=2|[[File:84.5° × 95.5° great rectangle.png|100px]] |rowspan=2|Great rectangles<br> in <big>☐</big> planes |rowspan=2| | |colspan=2| |rowspan=2|16₀<br><br>#7+1 |- style="background: gainsboro;"| |84.5~° |{{radic|0.81~}} |1.345~ |95.5~° |{{radic|2.19~}} |1.480~ |- style="background: gainsboro;"| |rowspan=2|15₀<br><br>#7 |<math>\pi / 2</math> |colspan=2| |rowspan=2|[[File:Great square rectangle.png|100px]] |rowspan=2|4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}} / 27<br> in 4050 <big>☐</big> planes |rowspan=2|4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |colspan=2| |rowspan=2|15₀<br><br>#7 |- style="background: gainsboro;"| |90° |{{radic|2}} |1.414~ |90° |{{radic|2}} |1.414~ |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons / 4".}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1, 2, & 7 are each overlapping pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hulls 6 & 8 are pairs of [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.]] {{Clear}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the length of the edge (the '''#1 [[#Chords|chord]]''' of the 120-cell) is {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270. Eight {{Color|orange}} vertices lie at the Cartesian coordinates (±φ³{{radic|8}}, ±φ³{{radic|8}}, ±φ³{{radic|8}}) relative to origin at the cell center. They form a cube (dashed lines) of edge length {{sfrac|1|φ{{radic|2}}}} ≈ 0.437 (the pentagon diagonal, and the '''#2 chord''' of the 120-cell). The face diagonals of the cube (not shown) of edge length {{sfrac|1|φ}} ≈ 0.618 are the edges of tetrahedral cells inscribed in the cube (600-cell edges, and the '''#3 chord''' of the 120-cell). The diameter of the dodecahedron is {{sfrac|{{radic|3}}|φ{{radic|2}}}} ≈ 0.757 (the cube diagonal, and the '''#4 chord''' of the 120-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer|Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer|Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an irregular dodecagon in a central plane.{{Efn|[[File:Great dodecagon of the 120-cell.png|thumb|200px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]]The 120-cell has an irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] of 6 edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} The irregular great dodecagon contains two irregular great hexagons ({{color|red|red}}) inscribed in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The twelve regular hexagon edges (#5 chords), the six cell-diameter edges of the dodecagon (#4 chords), and the six 120-cell edges of the dodecagon (#1 chords), are all chords of the same great circle, but the other 24 zig-zag edges (#1 chords, not shown) that bridge the six #4 edges of the dodecagon do not lie in this great circle plane. The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of dodecagon planes. The 120-cell contains 200 such {12} central planes (100 completely orthogonal pairs), the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in each of the 10 inscribed 600-cells.{{Efn|The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} All ten 600-cells occupy the same set of 200 irregular great dodecagon central planes.{{Efn|name=irregular great dodecagon}} There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint great hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells.|name=same 200 planes}}|name=irregular great dodecagon}} Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each irregular great dodecagon, in alternate positions.{{Efn|name=irregular great dodecagon}} === Perspective projections === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} As in all the illustrations in this article, only the edges of the 120-cell appear in these renderings. All the other [[#Chords|chords]] are not shown. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them. These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (below left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating animations above, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Orthogonal projections === [[W:Orthogonal projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} 3-dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image. {| class="wikitable" style="width:540px;" |+3D orthographic projections |[[File:120Cell 3D.png|270px]]<br>3D isometric projection |align=center|[[File:Cell120.ogv|270px]]<br>Animated 4D rotation |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell === The '''Davis 120-cell''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist}} ==Citations== {{Reflist}} ==References== {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book)}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd | isbn= }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] 4ku65wi649xstmhjpyupw7kz00v0vzv 0 264290 2691692 2691257 2024-12-12T22:00:53Z Scogdill 1331941 /* 1902 */ 2691692 wikitext text/x-wiki [[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] 1900s ==1900== 1900, early, [[Social Victorians/People/Mathers|MacGregor and Moina Mathers]] were living at 87 Rue Mozart, Paris (Howe 203). ===January 1900=== ====1 January 1900, Monday, New Year's Day==== ====13 January 1900, Tuesday==== <blockquote>THE HOUSEHOLD TROOPS. ENTERTAINMENT AT HER MAJESTY'S. The Prince and Princess of Wales, accompanied by Princess Victoria and Prince Charles of Denmark, attended the entertainment to aid the widows and orphans of her Majesty's Household Troops, organised by Mrs. Arthur Paget and presented under the direction of Mr. H. Beerbohm Tree at Her Majesty's Theatre last night. ... [The major part of this story is the program of the entertainment, in which [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], among others, played an important part.] Among those present at the entertainment were: The Prince and Princess of Wales, Princess Victoria of Wales, and Prince Charles of Denmark, the French Ambassador, the Russian Ambassador, the Portuguese Minister, Count Mensdorff, the Austrian Embassy, Prince and Princess Demidoff, Prince and Princess Hatzfeldt, Prince and Princess Alexis Dolgorouki, Count and Countess Roman Potocki, Count and Countess Alexander Münister, the Duke and Duchess of Devonshire, the Marquis of Downshire, the Earl and Countess of Cork, the Earl and Countess of Westmorland, the Earl and Countess of Gosford, the Earl of Lathom, the Countess of Ancaster, the Countess of Wilton, the Countess of Yarborough, the Countess of Huntingdon, Viscount Curzon, Lord and Lady Farquhar, Lord and Lady Savile, Lord Rowton, Lord Westbury, Baroness d'Erlanger, Count and Countess Seilern, Lord and Lady Ribblesdale, Lord and Lady Hothfield, Lord and Lady Raincliffe, Lord Wandsworth, Lord Charles Montagu, Lady Cunard, Sir Edgar and Lady Helen Vincent, Lady Kathleen and Mr. Pilkington, Lady Violet Brassey, Lady Grey Egerton, the Hon. Humphry and Lady Feodorowna Sturt, Lady Ripley, Lady Katherine Coke, Lady Agneta Montagu, Lady Tatton Sykes, Lady Templemore, Lady Florence Grant, Lady Garrick, Lady Pearson, Lady Constance Haddon, Sir F. Burdett, the Hon. M. Charteris, Sir A. de la Rue, Sir Frederick and Lady Milner, the Hon. E. Stonor, Sir Edward and Lady Sassoon, Mrs. Joseph Chamberlain, the Hon. Mrs. Lawrence, the Hon. Mrs. Napier, Sir Charles Forbes, Mrs. Bradley Martin, Mrs. Cornwallis West, Mr. Arnold Morley, Mr. L. Neumann, Madame Vagliano, Mr. Gillett, Mrs. Godfrey Samuelson, Mrs. Reginald Ward, Mr. and Mrs. Arthur Wilson, Mr. Menzies, Mr. Dreyfous [sic], Mrs. George Coats, Mr. Hartmann, Mrs. Rube, Mrs. Neumann, Mr. Lukach, Mrs. Candy, Mr. Bargrave Deane, Mr. L. V. Harcourt, Mrs. Oppenheim, Mrs. Lionel Phillips, Mr. King. Mr. James Finch, Mrs. Clayton Glyn, Miss Van Wart, Mr. Hall Walker, Mr. Drexell, Mrs. Van Raalte, Mr. Alfred Beit, Mr. Douglas Uzielli, Mrs. Alfred Harmsworth, Mr. Munday, Mrs. William James, Mrs. Newhouse, Mrs. Max Waechter, Mr. G. Prentis, Mrs. M'Calmont, Mr. Blacklock, Mrs. Ausell, Captain Holford (Equerry to the Prince of Wales), Mr. De Nino, Mrs. Keyser, Mrs. Fleming, Mrs. Breitmeyer, Mrs. Wernher, Mrs. Armour, Mr. Van Alan, Mrs. Ewart, Mrs. Carl Meyer, Mrs. Powell, Mr. Hambro, Colonel Charles Allen, Colonel Cunningham, Mrs.Hutchinson, Mrs. Schumacher, Colonel Kennard, Mrs. Fludyer, Mrs. Williamson, Mr. Thellusson, Mr. Sackville West, Captain M'Neil, Mrs. Dalrymple Hamilton, Mrs. Penn Curzon, Mrs. Hamar Bass, Mrs. Kuhliug, General Stracey, Mrs. Jeffcock, Colonel Thynne. (1900-02-14 Morning Post).</blockquote> ====17 January 1900, Saturday==== 1900 February 17, Lady Greville writes about the amateur theatricals Muriel Wilson is involved in: <blockquote>The most notable social event of the week was the amateur performance of tableaux at Her Majesty's Theatre. One is accustomed to the amateurs under every aspect, leaping in where angels fear to tread, essaying the most difficult parts, dabbling in the arts of music and literature, but so full and rich and interesting a performance has rarely been given before. To begin with, there was a masque, modelled on the Elizabethan lines, with song and dance, and special music composed for the occasion by Mr. Hamish McCunn, dresses statuesque and graceful, and a bevy of pretty women to carry out the idea. One original feature there was, too, which certainly did not present itself before our Virgin Queen, and that was the graceful fencing of Miss Lowther, who looked an ideal young champion in her russet suit and jaunty little cap. A very young debutante appeared in the person of Miss Viola Tree, who, dressed in the nest diaphanous garments, acted with a grace and lightness that promises well for her future career. Mrs. Crutchly, as "Glory," appeared amid a din of thunder and a rosy glare of limelight, and clashed her cymbals in truly determined fashion. An element of wildness suited to the character, distinguished her agreeable posturing, and her high spiked crown gave distinct individuality to the representation. Mrs. Martineau, Hebe-like in a white robe and a large crown of roses, as if she had just stepped out of a picture by Leighton, then danced and took the palm for poetry and suppleness of movement; Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], meanwhile, having daringly shot up through a trap-door in scarlet robes with a flaming torch, announced herself as "War," and beckoned to Glory, Victory, and Prosperity, when they finished their performance, to sit beside her on her throne. "Rumour," alias Mr. Gervase Cary Elwes, sang an excellent topical song, attired in a quaint garb covered with interrogations, and carrying an electric telegraph-post in her hand. Lady Maud Warrender, as "Pity," advanced from a barge that had just arrived, and sang a doleful ditty which made one wish "Pity" might combine a sense of gaiety. But as Mrs. Willie James, in the part of "Mercy," dressed as a nurse, recited some bright lines anent Tommy, to the accompaniment of distant fifes and drums, the audience decided to take this as a satisfactory compensation. All being now harmoniously arranged, "War" performed a sleight-of-hand feat, divested herself of her red dress, her headgear of flaming serpents, and her glistening breastpiece, and appeared in virgin white, crowned with roses, as “Peace," surrounded by “Music" in a gorgeous gown of gold tissue, by “Painting," “Science," and “Literature." A pleasant finaleof gay music brought the Masque to a close, and left a decidedly agreeable and novel impression behind it. Tableaux then followed, all more or less well grouped by well-known artists, and represented by beautiful women of Society. Among the familiar faces were Lady St. Oswald, Lady Mary Sackville, Miss Agatha Thynne, Mrs. Fitz Ponsonby, Lady Maitland, Madame von André, &c., but neither Lady Helen Vincent, Lady De Grey, Lady Cynthia Graham, the Duchess of Portland, nor many other well-known and lovely ladies took part in the performance. Finally, came the Patriotic Tableau, which had evidently engaged all the energies of the organisers of the fête. On a high throne, with a most realistic lion, open-mouthed and fierce-looking, beside her, sat Lady Westmoreland as "Great Britain," a stately and dignified figure in white satin, draped in a red cloak and crowned with a large wreath of laurel. The stage on each side was lined by genuine stalwart Guardsmen, and to the sound of lively martial music, composed and conducted by Sir Arthur Sullivan, slowly advanced a procession of Great Britain's dependencies, figured by ladies magnificently costumed, their long jewelled trains borne by two little pages in cloth of gold brocade coats, with black silk legs. Very beautiful were the blendings of the colours in this tableau, artistically designed by Mr. Percy Anderson. Lady Claude Hamilton, as "British Columbia," moved with stately gait in a robe of palest green; Lady Feo Sturt glittered barbarically with jewels; her headdress and her bosom were covered with gems. As the typical representative of "India," she was dressed in apricot colour and bore branches of hibiscus in her hands. Mrs. Hwfa Williams, in blazing red, carried a parrot and some red flowers. The Hon. Barbara Lister looked lovely and picturesque in her violet robes under a massive wreath of wisteria blossoms; Lady Raincliffe, wearing a curious high head-dress, was dressed in white to represent "Canada." "Rhodesia" made one of the prettiest figures in her khaki gown and cloak, with the coquettish hat and feathers and the red trimming associated with the Colonial Volunteers. "Natal" appeared appropriately clad all in black, while little "Nigeria," for the nonce, wore spotless white robes. / Miss Muriel Wilson spoke an ode, and looked striking in apricot and white, with a high diamond crown and a long standing-up white feather. None of the ladies suffered from shyness; they showed thorough acquaintance with the stage, and moved easily thereon. In fact, costumes, arrangements, music, and the glorious feast of beauty left nothing to be desired. The final impression in one's mind was that the stage produces strange effects. It idealises some faces, hardens others, and alters many. The large wreaths, almost grotesque in size, proved eminently becoming, and the Grecian draperies carried away the palm for beauty. After them our modern dress seems stiff, angular, and inartistic. The whole performance was one to be commended, and will no doubt be as successful financially as it was from the aesthetic and spectacular point of view. Mrs. James Stuart Wortley, who died last week, will be regretted by every class of society. This lady, a beauty in her youth, devoted the latter part of her life entirely to works of charity. She founded the East London Nursing Society, to the tender and skilful ministrations of which many a poor woman owes her return to health, and in every philanthropic scheme, emigration, the befriending of young servants, and the education of youth, she took a lively interest. Her clear sense, her logical grasp of subjects and her immense activity were of infinite service in everything she undertook, and her memory will smell sweet in the hearts of the many who loved and depended on her. I really wonder at the patience of the British taxpayer. During the snow of this week Belgravia, Eaton, and other fashionable squares, remained a morass of slush, ice, and half-melted snow. The pavements as slippery as glass had not been cleansed, and only at the risk of one's life one made one's way from street to street. (Greville 7, Col. 1a-2a)</blockquote> '''25 January 1900, Thursday''' David Lindsay, [[Social Victorians/People/Crawford and Balcarres|Lord Balcarres]] and Constance Lilian Pelly married: <blockquote> MARRIAGE OF LORD BALCARRES. The marriage of Lord Balcarres, M.P. for North Lancashire, eldest son of the Earl of Crawford of Balcarres House, Fife, and Haigh Hall, Wigan, to Miss Pelly, daughter of the late Sir H. Peily, Bart., and granddaughter of the Earl of Wemyss, was solemnised yesterday (Thursday) at St Margaret's Church, Westminster, in the presence of a large gathering of friends. Among the invited guests were the Earl and Countess of Crawford, the Dowager Countess of Crawford, the Earl of Wemyss, Lord and Lady Elcho, the Hon. E. Lindsay, the Hon. Lionel Lindsay, the Hon. Ronald Lindsay, Lord and Lady Cowper, Mr. A. J. Balfour, the Hon. L. Greville, and many othsrs. The service was fully choral, and was conducted by the Bishop of Stepney, assisted by the the Rev. Canon Gore. Mr Yorke, the stepfather of the bride, gave her away. She wore a dress of white velvet, draped with old Brussels lace, the gift of the Dowager Countess of Crawford: chiffon veil and wreath of natural orange blossoms. Her only ornament was a Maltese cross of diamonds, also the gift of the Dowager Countess of Crawford. There were nine bridesmaids. Miss Pelly, sister of the bride) [sic], the Hon. Mary Vasey, the Hon. Cynthia Charteris, Miss Brodrick, Miss Sybil Brodrick, Miss Benita Pelly, the Hon. Aline Menjendie, Miss Daisy Benson, and Miss Madeline Bourke. They were attired alike in costumes of white de chine, with lace insertions, with blue chiffon hat, trimmed with plumes of white and blue ostrich feathers. They carried bouquets of violets, and wore red enamel brooches with diamond centres and pearl drops, the gifts of the bridegroom. The Hon E. Lindsay supported his brother as best man. At the conclusion of the ceremony the guests drove to the town residence of the bride's mother in Queen Anne's Gate, where the wedding reception was held. Later in the day the newly-married couple left town for Wrest Park, Ampthill, kindly lent them for the honeymoon by Earl and Countess Cowper. Princess Louise (the Marchioness of Lorne) sent the bride a handsome silver basket as a wedding present.<ref>"Marriage of Lord Balcarres." ''Dundee Courier'' 26 January 1900 Friday: 4 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000164/19000126/105/0004.</ref> </blockquote> ===February=== 1900, February, a brief account of the Matherses' Isis ceremony appeared in "the New York periodical the ''Humanist'', February 1900" (Howe 201). ==== 15 February 1900, Thursday ==== A number of familiar people took part in [[Social Victorians/People/Arthur Stanley Wilson|Enid Wilson]]'s wedding to the Earl of Chesterfield:<blockquote>This afternoon, at St. Mark's Church, North Audley-street, the [[Social Victorians/People/Chesterfield|Earl of Chesterfield]] is to be married to Miss Enid Wilson, second daughter of Mr. Charles Wilson, M.P., of Marter Priory, Yorkshire, and 41, Grosvenor-square. The bride, who will be given away by her father, will wear a dress of white crêpe de chine wrought with silver and trimmed with fine old lace and ermin. The bridesmaids will be Miss Gladys Wilson, sister of the bride, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], her cousin, Lady Aldra Acheson, daughter of the Earl [[Social Victorians/People/Gosford|and Countess of Gosford]], Lady Mary Willoughby, daughter of the [[Social Victorians/People/Ancaster|Earl and Countess of Ancastor]], Lady Marjorie Carrington, daughter of [[Social Victorians/People/Carrington|Earl and Countess Carrington]], Miss Daphne Bourke, the four-year-old daughter of the [[Social Victorians/People/Bourke|Hon. Algernon and Mrs. Bourke]], [[Social Victorians/People/Balfour|Miss Balfour]], and [[Social Victorians/People/Paget Family|Miss Paget]]. Lace Empire dresses and long bright red cloth Directoire coats trimmed with sable and hats to match. They will carry sable muffs, the gifts to them of the bridegroom. Viscount Ednam, the [[Social Victorians/People/Dudley|Earl and Countess of Dudley]]'s only son, aged six, and Lord Wendover, the only son of Earl and Countess Carrington, aged something over four, will be the youthful trainbearers. After the wedding Mrs. Charles Wilson will hold a reception at 41, Grosvenor-square. The Prince of Wales has given Lord Chesterfield a remarkably handsome embossed silver cigarbox, lined with cedar, monogrammed and coroneted, accompanied by a letter written by his Royal Highness to the bridegroom, cordially wishing him every happiness. The Duke of Fife's gift is a cedar-lined plain silver cigarette-case.<ref>"London Day by Day." ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph & Courier'' (London). Print p. 8.</ref></blockquote>Another, more local report: <blockquote>There was very large and fashionable assembly at St. Mark’s Church, North Audley-street, London, on Thursday afternoon, to witness the very pretty wedding of the Right Hon. the Earl of Chesterfield, P.C., of Holme Lacey, Hereford, and Miss Enid Wilson, fourth daughter of Mr. Charles H. Wilson, M.P. for Hull, of Warter Priory, York, and 41, Grosvenor-square, London. The service was fully choral, and the church handsomely decorated. There were seven bridesmaids in attendance upon the bride. These young ladies were Miss Gladys Wilson (sister). Miss Muriel Wilson (cousin of the bride). Lady Mary Willoughby, Lady Alexandra Acheson, Lady Marjorie Carrington, Miss Dorothy Paget, and Miss Alice Balfour, who were in costumes of quaint, old-fashioned riding coats of red cloth, with white muslin skirts. The local guests included Sir James and Lady Reckitt, Sir James and Lady Woodhouse, Lord and Lady Herries, Mr. Philip Hodgson, Lord and Lady Raincliffe, Mr. and Mrs. Kenneth Wilson, Mr. and Mrs. Stanley Wilson, Mr. and Mrs. Strickland Constable. Presents wore also received from Commander Bethell (silver candlestick). Hr. and Mrs. George A. Duncombe, Beverley (Louis XV. sofa). Mr. and Mrs. Frank Lambert, Beverley (inlaid writing table), Colonel and Mrs. Goddard, Cottingham (silver mirror), Mr. Haslewood Taylor, Beverley (pair of prints). (1900-02-21 Beverley Echo)</blockquote> ==== 27 February, 1900, Tuesday ==== Mardi Gras ===April 1900=== ==== 8 April 1900, Sunday ==== Palm Sunday ====14 April 1900, Saturday==== Wynn Westcott assumed W. A. Ayton was on, as he wrote, "the Committee to investigate the G. D. which contains Yeats, Bullock and I suppose Ayton" (Howe 217). ====20 April 1900, Friday==== The R.R. et A.C. was code named Research and Archaeological Association (Howe 226) ====21 April 1900, Saturday==== The Inner Order of the Golden Dawn met at 116 Netherwood Road, West Kensington (Howe 227). ===May 1900=== ====26 May 1900, Saturday==== Arthur Sullivan is visited by "Sir George Martin, the organist at St. Paul's Cathedral, and Colonel Arthur Collins, one of the royal equerries" to get him to write a Te Deum thanking God for the end of the Boer War (Ainger, Michael. Gilbert and Sullivan: a Dual Biography. P. 381.). ====30 May 1900, Wednesday==== Derby Day. According to the Morning Post, <quote>The Derby Day. / The Archbishops of Canterbury and York hold a Reception of Colonial and Missionary Church Workers in the Great Hall of the Church House, 4.30 to 6.30. / ... May Fair and Bazaar, St. George's Drill Hall, Davies-street, Berkeley-square, opened by Lady Edward Spencer Churchill, 2.30.</quote> ("Arrangements for This Day." The Morning Post Wednesday, 30 May 1900: p. 7 [of 12], Col. 6C) ===June 1900=== Summer 1900: WBY summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ==== 3 June 1900, Sunday ==== Whit Sunday (Pentecost) Whitsun party at Sandringham House, described by Lord Knutsford in his letters and summarized by Anita Leslie, whose parent's generation remembered some of these people Knutsford mentions as present: * The Prince and Princess of Wales * Princess Victoria * Other daughters of the Prince and Princess of Wales * Lord Knutsford * [[Social Victorians/People/Ripon|Lord and Lady Gladys de Grey]] * Luís De Soveral * Tosti * [[Social Victorians/People/Durham|Hon. George Lambton]] * [[Social Victorians/People/Churchill|Lady Randolph Churchill]] * [[Social Victorians/People/Holford|Holford]] * Lady Musgrave Leslie's summary of Knutsford's letters:<blockquote>The Whitsun party that year included Lord and Lady de Grey, De Soveral, whose caustic wit always lightened Edward's humour, Tosti, the famous baritone-songwriter (Alexandra and her daughters were so musical — strumming away ''à quatre mains'' while Totti's voice made chandeliers vibrate in after-dinner songs), the Hon. George Lambton (racing trainer), and Lady Randolph Churchill, "just back from her hospital ship which had been a boon in South Africa, but fractiously insisting she is going to marry George Cornwallis-West." Lord Knutsford describes the chattering guests travelling in that special train coach from St. Pancras to Wolverton Station where the house party was met by royal carriages with officious flunkeys in red livery who dealt with the luggage — and ''such'' luggage! Big trunks had to be brought for a few days' stay so that the correct attire could be produced for every meal and outing. How exciting to drive through a forest of rhododendrons and to disembark in front of Sandringham House. The royal host and hostess stood in the hall to welcome their guests. After handshakes Queen Alexandra sat down to pour tea. Dinner was at 9 <small>P</small>.<small>M</small>. (at Sandringham all clocks were kept half an hour ahead of time). Footmen informed the gentlemen what waistcoats were to be worn. Ladies' maids scurried to the ironing rooms. At nine, having assembled in the drawing room, each man was told whom he must escort into dinner and where to sit. This saved hesitation and embarrassment. On this occasion Knutsford describes the Prince giving his arm to Lady de Grey, while Alexandra walked beside De Soveral and Lord de Grey escorted the unmarried Princess Victoria. There were, of course, no cocktails, but exquisite wines accompanied each course. The Prince never drank more than a glass or so of claret at dinner and a brandy after the last course. When the ladies left the dining room cigarettes and cigars were brought by footmen. Heavy drinking was never encouraged, and / after half an hour the gentlemen moved to the drawing room to chat with the ladies, until Alexandra rose and they retired to their bedrooms where the ladies' maids would be waiting to unlace them from their gorgeous satin and velvet gowns. Hard as the existence of a servant might be, they were perhaps consoled by the colossal meals offered in recompense for late hours. A five-course breakfast could be consumed by every scullery maid if she so desired, and many a working-class mother strove to "get her daughter's knees under a good table." When the ladies had disappeared upstairs the men went to the billiards room, where the Prince, who idolised his dogs, would roar with laughter when his black bulldog nipped the legs of players. No one could go to bed before Edward, but at twelve-thirty he would certainly retire. There was no thought of any hanky-panky after hours at Sandringham. That would have been considered bad taste and an insult to the royal hostess. On Sunday morning the breakfast gong sounded at 10 <small>A</small>.<small>M</small>. Then came church and a stroll in the garden until lunch at one-thirty. After a fairly heavy meal the ladies went upstairs to change into walking skirts and strong boots. The whole party then underwent a slow three-hour walk to the kennels and stables and farm. Talk was almost entirely about animals — dogs, pedigree cattle and, of course, race horses. Knutsford noticed Alexandra's "touching girl-like love" for every stone and corner of Sandrringham. She reminded him of "a bird escaped from a cage." Certainly the royal pair were never so happy as in this big Norfolk house, which they regarded as home, but guests grew weary of trying to do the right thing. Knutsford found dinner very wearing, with the conversation in mingled English and French: "they drop from one to another in the same sentence." Then came the local Whitsunday sports. Off drove the house party — Lady de Grey and Holford in the first carriage with Edward. Knutsford found himself in the second carriage with Princess Victoria and Lady Randolph Churchill and Lady Musgrave. The ladies wore coloured blouses and contrasting skirts and jackets over their blouses, white gloves and feather boas. A brisk wind nearly blew off their huge hats. Lady Musgrave in particular had difficulty with her concoction. "Send it to the bazaar!" cried Alexandra, and everyone roared with laughter. Sandringham parties were called "informal," but what a relief, nevertheless, when they all got back to the station in those regal carriages followed by the four horse-drawn vans of luggage. In this spring of 1900 the visitors departed to their homes full to / the brim of food and anecdote. Jennie, who had been argumentative all weekend, would almost immediately marry her young George. Gladys de Grey would get on her newly installed phone to admirer number one, the Hon. Reginald Listen, or if he was not available to admirer number two, Sir John Listen-Kaye. Ladies were now able to ring the men up and guardedly converse instead of sending dangerous notes. Servants might overhear but there would be nothing ''on paper''.<ref>Leslie, Anita. ''The Marlborough House Set''. Doubleday, 1973.</ref>{{rp|195–197}}</blockquote> ====26 June 1900, Tuesday==== There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. ==== 28 June 1900, Thursday ==== Lady Randolph Churchill and George Cornwallis-West married at St. Paul's, Knightsbridge.<ref>Martin, Ralph G. ''Lady Randolph Churchill : A Biography''. Cardinal, 1974. Internet Archive: https://archive.org/details/ladyrandolphchur0002mart_w8p2/.</ref>{{rp|220–223}} ===July 1900=== ====27 July 1900, Friday==== The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] had dinner at the Arthur Wilsons’:<blockquote>[[Social Victorians/People/Arthur Stanley Wilson|Mr and Mrs Arthur Wilson]] were honoured with the presence of the Prince of Wales at dinner on Friday night. Amongst the guests were the Portuguese Minister, Count Mensdorff, Duke of Roxburghe, Lady Georgina Curzon, Captain and Lady Sarah Wilson (arrived that morning from South Africa), Lord and Lady Tweedmouth, Lord Herbert Vane Tempest, Viscount Villiers, Lady Norreys, Lady Gerard, [[Social Victorians/People/Keppel|Hon Mrs Keppel]], Sir Edward and Lady Colebrook, Mr and Mrs Grenfell, Lady Lister Kaye, Mrs Arthur Paget, Mr and Mrs Arthur Sassoon, Hon. W. Erskine, Mr and Mrs J. Menzies, General Oliphant, Miss Jane Thornewell, Mrs Kenneth Wilson, and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]]. (1900-07-30 Hull Daily Mail)</blockquote> ===October 1900=== ====31 October 1900, Wednesday==== Halloween. ===November 1900=== ====5 November 1900, Monday==== Guy Fawkes Day ====9 November 1900, Friday==== A debutante dance for Miss Helyar: <quote>In honour of the coming of age of Miss Helyar, a small dance was given by Lady Savile, at Rufford Abbey, last night. The number of invitations was not so large as it would have been but for the war. The house party included Mrs. and Miss Cavendish Bentinck, Lady Juliet Lowther, Lady Evelyn Ward, Lady Mabel Crichton, Mrs Kenneth Wilson, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Sir Berkeley Sheffield, Miss Sheffield, Lord Hyde, Lord Herbert, the Hon. B. Ward, the Hon. E. FitzGerald, the Hon. W. Erskine, Mr. Laycock, Captain Brinton, the Hon. George Peel, Mr. Harris, Captain Tharp, Captain Heneage, and the Hon. G. Portman.</quote> (1900-11-10 Yorkshire Post) ====27 November 1900, Tuesday==== Arthur Sullivan's funeral: <quote>At eleven o'clock on Tuesday, November 27th, the [366/367] funeral procession set forth from Victoria Street, Westminster, on its mournful way, first to the Chapel Royal, St. James's, where, by command of the Queen, part of the Burial Service was to take place, and thence to St. Paul's. Throughout the line of route flags drooped at half-mast, whilst beneath them people crowded in their thousands, bare-headed and in silence, waiting to pay their last tribute of respect and gratitude to the lamented master whose genius had done so much to brighten their lives for the past five-and-twenty years. [new paragraph] Into the Royal Chapel, where Arthur Sullivan had begun his career as a chorister, was borne the casket containing his remains. On either side stood men and women famous in society and the wider world of Art in all its branches. The Queen was represented by Sir Walter Parratt, Master of Music, who was the bearer of a wreath with the inscription: "A mark of sincere admiration for his musical talents from Queen Victoria." Sir Hubert Parry represented the Prince of Wales; the German Emperor was represented by Prince Lynar, Attache of the German Embassy; Prince and Princess Christian by Colonel the Hon. Charles Eliot, and the Duke of Cambridge by General Bateson. Among the congregation at the Chapel Royal were seen the United States Ambassador; the Earl and Countess of Strafford; Theresa, Countess of Shrewsbury; the Countess of Essex; Lord Glenesk; Lord Rowton; Lord Crofton; Lady Catherine Coke; the Dean of Westminster; Lady Bancroft; Lady [367/368] Barnby; Mr. Arthur Chappell; Mr. and Mrs. F. C. Burnand; Mr. Arthur W. Pinero; Mr. Haddon Chambers; Lieutenant Dan Godfrey; Signor Tosti; Mr. George Grossmith; Mr. Rutland Barrington; Miss Macintyre; Mrs. Ronalds; Canon Duckworth; Lady Lewis; Miss Ella Russell; Mr. Augustus Manns; Mr. Charles Wyndham; Captain Basil Hood; the Chairman and Secretary of Leeds Musical Festival; and Representatives of various British Musical Associations. The Pall-bearers were Sir Squire Bancroft, Mr. Francois Cellier, Colonel A. Collins (one of the Royal Equerries), Sir Frederick Bridge, Sir George Lewis, Sir Alexander Mackenzie, Sir George Martin, and Sir John Stainer. [new paragraph] he chief mourners were Mr. Herbert Sullivan (nephew), Mr. John Sullivan (uncle), Mrs. Holmes, and Miss Jane Sullivan (nieces), Mr. Wilfred Bendall (Sullivan's secretary), Mr. B. W. Findon, Mr. Edward Dicey, Mr. C. W. Mathews, Mrs. D'Oyly Carte, Dr. Buxton Browne, Mr. Arthur Wagg, Mr. Fred Walker, Mr. Dreseden and Sir Arthur's servants. [new paragraph] Much to their regret, neither Mr. Gilbert nor Mr. Carte was able to attend the funeral. The first was on the Continent for the benefit of his health, the second was laid up by serious illness. The present writer also, having been absent from London at the time, has not the advantage of an eye-witness to give a graphic description of the funeral obsequies of his old friend; and so, rather than attempt to paint the picture from imagination, he gladly avails himself [368/369] again of the courtesy of his brother-author who is so generous as to lend the aid of his experience. [new paragraph] In these sympathetic words, Mr. Findon describes the scenes and incidents in which, as a chief mourner, he took part at the Chapel Royal and St. Paul's Cathedral: <blockquote>". . . As the casket was borne into the Chapel, it was impossible to avoid thinking of those days when Sullivan himself had worn the gold and scarlet coat of a Chapel Royal Chorister, and his sweet young voice had rung through the sacred edifice. Then the world and its honours lay before him, but we doubt if even in the most sanguine moments of impulsive boyhood he imagined the greatness that one day would be his, or that his bier would pass within those honoured walls amid the silent demonstration of a mourning people. The anthem, 'Yea, though I walk through the valley of the shadow of death,' from his oratorio 'The Light of the World,' was beautifully sung, and the pathos of the music bathed many a face in tears, and touched a tender spot in more than one loving heart. Another of the dead master's exquisite thoughts, ' Wreaths for our graves the Lord has given,' brought the Service at the Chapel Royal to an end, and the procession passed on its way to St. Paul's Cathedral, which was crowded with sympathetic spectators. "Clerical etiquette and cathedral dignity compelled the beginning of the Burial Service anew, and when the coffin had been lowered into the crypt there came the most poignant moment of the long ceremonial. [new paragraph] "Close to the open vault sat the members of the Savoy Opera Company, including his life-long friend, Mr. Francois Cellier, who had been associated as chef d'orchestre with all his comic operas, and, after [369/370] the Benediction had been given, they sang in voices charged with emotion the touching chorus, 'Brother, thou art gone before us,' from ' The Martyr of Antioch.' The effect was quite remarkable, inasmuch as it was one of those incidents which come but rarely in a life-time."</blockquote>It was not in London alone that people mourned for Arthur Sullivan on that November day. Throughout Great Britain and Ireland, on the Continent of Europe, in America and farther across the seas, thousands of fond and grateful hearts ached with grief at the thought that England's dear master of melody had passed away into the silent land. From high-born personages and from people of low estate came floral emblems, wreaths, crosses, and lyres innumerable. Conspicuous among them was a beautiful harp of purple blossoms with strings — one broken — of white violets. To this offering was attached a card bearing the inscription:<blockquote>In Memoriam ARTHUR SEYMOUR SULLIVAN Born 13 May, 1842. Died 22 Nov., 1900 FROM MR. D'OYLY CARTE'S "ROSE OF PERSIA" TOURING COMPANY IN TOKEN OF THEIR AFFECTIONATE REGARD <poem>Dear Master, since thy magic harp is broken, Where shall we find new melodies^ to sing? The grief we feel may not in words be spoken; Our voices with thy songs now heav'nward wing. Whilst on thy tomb we lay this humble token Of love which to thy memory shall cling.</poem> BELFAST, 24th November, 1900.</blockquote> [370/371] These simple lines but half expressed the love and esteem in which Sir Arthur Sullivan was held by all whose privilege it was to have been associated with him, and to have served, however humbly, his proud and brilliant life-cause.</quote> (Cellier, François, and Cunningham Bridgeman. Gilbert and Sullivan and their operas: with recollections and anecdotes of D. Pp. 366-371. Google Books: http://books.google.com/books?id=Au05AAAAIAAJ.) ====30 November 1900, Friday==== The wedding between Lady Randolph Churchill and George Cornwallis West at St. Paul's, Knightsbridge, occurred about this time. [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] attended, as did much of Society (1900-07-30 Times). ===December 1900=== ===25 December 1900, Tuesday==== Christmas Day ====26 December 1900, Wednesday==== Boxing Day ===Works Cited=== *[1900-02-14 Morning Post] "The Household Troops. Entertainment at Her Majesty's." Morning Post 14 February 1900, Wednesday: 3 [of 10], Col. 1a–2b [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19000214/014/0003 (accessed February 2020). *[1900-07-30 Hull Daily Mail] "Social Record." Hull Daily Mail 30 July 1900, Monday: 2 [of 6], Col. 5a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/19000730/007/0002 (accessed July 2019). *[1900-07-30 Times] "Court Circular." Times, 30 July 1900, p. 6. The Times Digital Archive, http://tinyurl.galegroup.com/tinyurl/AHR8r5. Accessed 20 June 2019. *[1900-11-10 Yorkshire Post] "Court and Personal." Yorkshire Post 10 November 1900, Saturday: 6 [of 14], Col. 4c [of 8]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000687/19001110/099/0006 (accessed July 2019). *Greville, Lady Violet. "Place aux Dames." The Graphic 17 February 1900, Saturday: 7 [of 40], Col.1a–2a, 2c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/19000217/008/0007 (accessed July 2019). [Col. 2c only for the last 2 paragraphs, not really relevant to Muriel Wilson] ==1901== ===January=== "There were no winter performances of opera at Covent Garden in those times: there was, in 1901, only a summer season" (Baring-Gould II 704, n. 14, quoting Rolfe Boswell). ====1 January 1901, Tuesday, New Year's Day==== ====16 January 1901, Wednesday==== Arnold Dolmetsch sent out notices that he was moving to 85 Charlotte Street, Fitzroy Square (Campbell 137-38). ====22 January 1901, Tuesday==== Queen Victoria died at Osborne House, on the Isle of Wight. ====23 January 1901, Wednesday==== Edward VII formally proclaimed “King of Great Britain and Ireland and Emperor of India, Defender of the Faith” "at Temple Bar, on St. Paul's Cathedral steps and at the Royal Exchange." "The Privy Council met in St. James' Palace at 2 o'clock in the afternoon for the purpose of signing the accession proclamation of Edward VII. The attendance at the meeting of the Council was more than 200." (Merrill, Arthur Lawrence, and Henry Davenport Northrop. Life and Times of Queen Victoria: Containing a Full Account of the Most Illustrious Reign of Any Soveriegn in the History of the World, Including the Early Life of Victoria; Her Accession to the Throne and Coronation; Marriage to Prince Albert; Great Events During Her Brilliant Reign; Personal Traits and Characteristics That Endeared Her to Her People; Graphic Descriptions of Her Charming Home Life; Noble Qualities as Wife and Mother; Royal Castles; Public Receptions; Wonderful Growth of the British Empire, Etc. To Which is Added the Life of King Edward VII., and Sketches of the Members of the Royal Family. Philadelphia, PA: World Bible House, 1901. Page 437. Google Books: http://books.google.com/books?id=Kx48AQAAIAAJ) ====26 January 1901, Saturday==== Arnold Dolmetsch gave a performance at his new domicile at 85 Charlotte Street, Fitzroy Square (Campbell 137-38). ===February 1901=== ====2 February 1901, Saturday==== Queen Victoria’s funeral at St. George’s Chapel, Windsor Chapel. Consuelo (Vanderbilt), Duchess of Marlborough was there: <blockquote>The service itself was magnificent. The stalls of the Knights of the Garter were occupied by the German Emperor and a dazzling array of kings, queens, ambassadors extraordinary, Indian princes, Colonial dignitaries, generals, admirals and courtiers. Consuelo wore the prescribed deep black mourning and crepe veil, which rather suited her, and it had the effect of extracting what she describes as a 'rare compliment' from her husband who remarked: 'If I die, I see you will not remain a widow long' — a conceit which suggests that he was more of his father's son than he cared to acknowledge. Consuelo later reflected that the funeral of Queen Victoria was a moment when it truly appeared that no other country in the world had an aristocrac so magnificent, nor a civil service so dedicated, which is precisely what was intended. The great doors were flung open as the royal cortege mounted the steps, a boom of distant guns and clanging swords the only sound other than the funeral march, until Margot Asquith broke the reverential silence with a quip. Consuelo thoroughly enjoyed herself at the reception in the Waterloo Chamber afterwards too. (Stuart, Amanda Mackenzie. Consuelo and Alva Vanderbilt: The Story of a Daughter and a Mother in the Gilded Age. New York and London: HarperCollins, 1005. Page 228. Google Books: http://books.google.com/books?id=44mhoIv12rEC)</blockquote> Also Henry James saw the funeral procession. ====3 February 1901, Sunday==== 1901 February 2–4?: Queen Victoria lay in state for 2 days between her funeral and her interment. ====4 February 1901, Monday==== Queen Victoria’s interment at Frogmore Mausoleum, Windsor Great Park. ====23 February 1901, Saturday==== The wedding of Hugh Richard Arthur, 2nd Duke of Westminster and Constance Edwina Cornwallis-West (1901-02-23 Cheshire Observer). ===March 1901=== Sometime in March 1901 Arthur Conan Doyle and Fletcher Robinson "were on a golfing holiday at the Royal Links Hotel at Cromer in Norfolk," where Robinson told Doyle a Dartmoor legend of "a spectral hound" (Baring-Gould II 113). Doyle's "The Hound of the Baskervilles" began publication in the ''Strand'' in January 1902. ===April 1901=== ====18-20 April 1901, Thursday-Saturday==== [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] and Mrs. Beerbohm Tree took part in 3 performances of <quote>Masks and Faces. The matinées have been organized by [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]], of Tranby Croft, in aid of the local fund of the Soldiers’ and Sailors’ Families Association. It was originally intended that the matinées should have been given in January last, but, owing to the death of Queen Victoria, they were postponed until Thursday, Friday, and Saturday last week. Additional interest was centered in the event, owing to the cast including no less a name than that of Mrs. Beerbohm Tree, while the fact that Miss Muriel Wilson was to appear as Peg Woffington aroused expectation.</quote> (1901-04-25 Stage) ===June 1901=== Summer 1901: William B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ====17 June 1901, Monday==== <quote>The "Women Writers" held their dinner at the Criterion on Monday, the 17th. Now Mr. Stephen Gwynn, in his paper entitled "A Theory of Talk," roundly asserts that women are less amusing than men. He says that there is no reason in nature why they should be, but that their inferiority is obvious. He points out that "thirty or forty men will meet at seven o'clock, dine together, and pass the evening very agreeably till midnight. Imagine thirty or forty women called upon to do the same; would they be able to amuse themselves?" It seems almost a pity that the exclusiveness of the women writers would not allow Mr. Gwynn personally to observe whether they were amused or bored on Monday night. In number there were nearly two hundred, and there certainly did not appear to be any lack of enjoyment or of laughter, but then it is also a fundamental belief with men that women are early adepts at hiding their true feelings. / Lucas Malet occupied the chair, and her carefully prepared speech was read out by Miss Sydney Phelps. Standing at the base of the statue of one of the world's greatest authors, and that, we regret to say, not a woman but a "mere man," Miss Phelps, speaking for Lucas Malet, said there was good cause for women to congratulate themselves that, whereas there had been Thackeray, Dickens, the brothers Kingsley, and Wilkie Collins among authors, authoresses could boast of George Eliot, Mrs. Gaskell, [33 Col B / 34 Col A] Miss Yonge, &c, and she felt that in the future they might equal, she would not say rival, their "brother man." At this courageous vaunt our glance involuntarily strayed to the statue, anticipating that it would be moved to at least a wink; but overwhelmed, perhaps, by the presence of so many "sisterwomen," it gave no sign. The speech was long, lasting for over thirty minutes. It touched on the evils of lowering work to what might be a present commercial but fleeting value; it contained much that was excellent, and tendered some good sound advice; perhaps it dwelt a trifle too insistently upon the obvious, and it was serious even to solemnity. But then "women are so serious." / Mme. Sarah Grand's reply was couched in far lighter vein. It slipped into the anecdotal, and was altogether more in the masculine line of after-dinner speaking. It offered no advice save on the advisability of laughter; it lingered for a moment on the sorrows of misinterpretation and misunderstanding, and included some amusing examples. Mme. Sarah Grand possesses a sympathetic voice, and is very pleasant to listen to. / It is characteristic of the gravity with which even in play hours women regard their "work" that the majority of guests preferred the more serious matter of Lucas Malet to the light personal note of Mme. Grand. The dinner itself was very good, and it was noticeable that whilst at the Authors' dinner on May 1 but few women availed themselves of the permission to smoke, at the women's function scarcely one was without a cigarette. Coffee was served at the table, and afterwards the company broke up into groups. / The committee numbered among its members Miss Beatrice Harraden, Mrs. Steel, Mrs. Craigie, Miss Christabel Coleridge, Miss Violet Hunt, and many other favourite writers. In the company present there were Dr. Jex-Blake, Mrs. Ady, Dr. Margaret Todd, Miss Adeline Sergeant, Mrs. Mona Caird, Mrs. Burnett-Smith, Mme. Albanesi, Miss Nora Maris, Miss Kenealy, and others; and the following presided at the tables : Lucas Malet, Mme. Sarah Grand, Mrs. de la Pasture, Miss Montresor, the Lady Mayoress, Mrs. L. T. Meade, Mrs. Alec Tweedie, Mrs. Walford, Mrs. B. M. Croker, Miss Violet Hunt, Miss Beatrice Harraden, Mrs. Belloc Lowndes, Miss Violet Brooke-Hunt, Miss Thorneycroft Fowler.</quote> ("The Women Writers' Dinner." The Author. Vol. XII, No. 2. 1 July 1901. Pp. 33–34.) ====26 June 1901, Wednesday==== There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. ====29 June 1901, Saturday==== "To-day sees the public inauguration of the Horniman Musem at Forest Hill. This collection of marvels from many lands, gathered together by a member of the Horniman family, has been generously presented to the public and housed in a handsome new building — set in the midst of fifteen acres, which are now dedicated to use as a public park. The entrance to the museum will be free." ("The Horniman Museum." Illustrated London News (London, England), Saturday, June 29, 1901; pg. 928; Issue 3245, Col. B) ===July 1901=== ====19 July 1901, Friday==== [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] hosted a concert at the Wilson house in Grosvenor-place in London: <quote>Mr. and Mrs. Arthur Wilson lent their house in Grosvenor-place on Friday afternoon for Miss Gwendoline Brogden’s concert. Miss Brogden, who is only eleven years old, is quite a prodigy. She sings quite exquisitely, and great many people, including Lady de Grey and Mrs. Arthur Wilson, are much interested in her future, which promises to be a very brilliant one. Lady Maud Warrender, Miss Rosamond Tufton, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. Bernard Ralt, Signor Ancona, and Signor Tosti, all promised to assist at the concert, and the tickets were a guinea each.</quote> (1901-07-24 Beverley Echo) ==== 25 July 1901, Thursday, 2:30 p.m. ==== The wedding of William Dixon Mann Thomson — Captain Mann Thomson in the Life Guards — and Violet Hemsley Duncan. Captain Mann Thomson's father had died in 1899. (Guests' names with their gifts set as an unordered list here, to save space; it was typeset as a long list of paragraphs in the newspaper story.)<blockquote>MARRIAGE OF CAPTAIN MANN THOMSON AND MISS DUNCAN. The marriage of Captain Mann Thomson, Royal Horse Guards, and Miss Violet Duncan, eldest daughter of Mr. A. Lauderdale Duncan, Knossington Grange, Oakham, took place in St. Peter's Chnrch, Eaton-square, London, on Thursday, the inst., 2.30 p.m. The bride, who was given away her father, wore a dress of white satin, draped with white and old Brussels lace, wreath of orange blossoms, and tulle veil. Her ornaments were pearls. She was attended by seven bridesmaids, viz.: — Miss Adèle, Miss Marjory, and Miss Esmè Duncan, sisters; Miss Dorothy and Miss Sybil Thompson, cousins of the bride; Miss Villiers, cousin of the bridegroom; and Miss Joan Dawson. They wore dresses of the palest pink silk, covered with pink gauze, collars of white lace, and pale pink chiffon baby hats. The bride's train was carried by Miss Duncan, her youngest sister. The bridesmaids carried bouquets of pink carnations, and wore diamond brooches in the shape of a violet with sapphire centre, the gifts the bridegroom. A detachment of non-commissioned officers and men of the bridegroom's troop lined the aisle during the ceremony. The bridegroom was supported by the Earl Arran as best man. The officiating clergy were the Rev. Ravenscroft Stewart, Vicar of All Saints', Ennismore-gardens, the Rev. G. Tanner, Rector of St. Peter's, Knossington, Leicestershire, and the Rev. H. Trower. After the ceremony, a reception was held at 8, Rutland-gate, the residence of Mr. and Mrs. Lauderdale Duncan. Among those present were the Duke and Duchess of Westminster, Dowager Countess of Chesterfield, Sir William and Lady Houldsworth, the Hon. C. and Mrs. Stanhope, Miss Hay, Lord and Lady Eglinton, Lord and Lady Castlereagh, Lord Ernest St. Maur, Lord and Lady Pembroke, Mrs. Adair, Mrs. Mann Thomson, Miss Mann Thompson, Earl Arran, Lord Cecil Manners, Mrs. and Miss Wilton Phipps, and many others. Later, the bride and bridegroom left for Dover, ''en route'' for the Continent, where they will spend the honeymoon. The bride's travelling dress was of pale blue crepe-de-chine, and black hat. There were about five hundred gifts from relations and friends. The following is a list:— * Bridegroom to Bride — Large diamond spray * Mrs. Mann Thomson (mother of bridegroom) — Diamond ring, diamond and sapphire bangle, and cheque * Mr. Lauderdale (father of bride) — Diamond and sapphire necklace * Mrs. Duncan (mother of bride) — Silver-mounted travelling bag * Dowager Lady Hay (bride's aunt) — Silver tea service * Miss Mann Thomson (bridegroom's sister) — Brougham * Mr. and Mrs. Butler Duncan (uncle and aunt) — Gold-mounted claret jug * The Misses Jackson (bridegroom's aunts) — Silver plate * Mr. H. Mann Thomson (brother) — Silver-mounted portmanteau * Mr. Charles Hunt — Diamond and pearl brooch * Miss Adele Duncan — Gold match-box * The Earl Arran — Gold cigarette case * Mr. and Mrs. Lucas — Bracelet * Earl of Arran — Set of diamond and pearl studs * Capt. and Lady Riddell — Bracelet * Mrs. and Miss Wilton Phipps — Gold and ruby buckle * Hon. H. Stanhope, R.N. — Brilliant buckle * Mr. and Mrs. Pennington — Ruby necklace * Mr. A. Butler Duncan — Necklace (old design) * Mr. and Mrs. Gervase Beckett — Sleeve links * Duke and Duchess of Westminster—Pair of silver candlesticks * Duchess of Roxburgh—Dresden china coffee service * The Countess of Shaftesbury — Walking-stick * The Earl of Arran — Umbrella * Lady Napier Magdala — Snuff-box * Sir Richard Waldie Griffith — Fan * Officers of the Royal Horse Guards — Massive silver vase * Lady Houldsworth — Silver inkstand * Viscount Ingestre — Silver waiter * Miss Hay — Silver coffee pot * Lady Hay — Silver tea caddy * The Countess of Chesterfield — Silver and brilliant-mounted photo frame * Lord Manners — Set four silver candlesticks * Lord and Lady Eglinton — Silver cigarette box * Earl and Countess of Ancaster — Pair of silver peppers * Lady Augusta Noel — Book-slide * Mr. and Mrs. Bradley-Martin — Old china coffee service in case * Mr. and Lady Wilfred Renshaw — Leather-covered book, "Where It?" * Mrs. Duncan — Silver-mounted stationery case and blotter * Sir Arthur Fludyer — Hunting crop * Lady Katherine Cole — Walking-stick * Lord Hamilton — Oak card table * Sir John Kelk — Writing case * Capt. Hon. E. St. Aubyn — Set of silver spoons in case * Capt. and Mrs. Burns-Hartopp — Set of silver asparagus tongs in case * Capt. Trotter — Silver sealing-wax stand * Capt. E. W. Clowes — Silver tobacco box * Mr. and Mrs. Sands Clayton — Silver scent bottle * Mr. and Mrs. John Hunt Clayton — Thermometer in silver-mounted case * Mr. and Mrs. Evan Hanbury — Clock * Major Atherley — Cigarette box * Mr. and Mrs. Richard Tryon — Card case * Mr. and Mrs. Hamilton Stubber — Table mirror in silver frame * Mr. and Mrs. Gretton — Pair of silver candlesticks * Miss Adele Duncan — Silver tea service * Hon. G. Crichton — Silver-mounted paper-knife * Mrs. Norman Lampson — Parasol * Capt. Gregson — Photo, "Guards at Pretoria" * Mr. Alfred Keyser — Leather bag * Mr. and Mrs. Armytage — lvory paper knife * Mrs. Boyce — Leather tray with two painted china plaques * Mr. and Mrs. A. B. Norman — Silver-mounted paper knife * The Master of Elibank — Pair of silver ash trays * Mr. Adrian Rose — Pair of silver toast racks * Mr. Archibald Smith — Hunting crop * Major Bradford Atkinson — Walking-stick * Mr. and Mrs. Stanhope — Painted china tea service * Mr. G. A. Grant — Stationery case * Mrs. Charles Inge — Copper and brass jardiniere * Col. and Mrs. Makins — Hunting crop * Mr. G. F. Trotter — Walking stick * Mr. and Misses Cardwell — Fan * Mrs. Dana — Thermometer * Mrs. Nugent — Card case * Mr. and Mrs. Ovey — Tortoiseshell box * Mr. F. Peake — Writing table * Capt. Boyce — Embroidered table cover * Mrs. Duncan — Dressing bag case * Mr. F. C. Fardell and Miss Gilbert Day — Brocaded satin cushion * Mr. and Mrs. Niel Robson — Visiting book * Mrs. R. B. Hay — Silver salts in case * Mr. and Mrs. Harold Broadbent — Pair silver peppers in case * —— Set silver knives in case * Mr. and Mrs. Greville Clayton — Six silver vases in case * Mr. and Mrs. Reginald H. Lewis — Pair silver peppers * Lord Ernest St. Maur — Set four silver fruit spoons in case * Rev. Geo. and Mrs. Tanner — Pair of silver salts * Capt. Thomson's Valet and Groom — Pair of silver peppers * Mr. Alick Duncan — Silver jug * Mr. and Mrs. A. Brocklehurst — Silver timepiece in case * Lieut.-Col. and Mrs. Blackburn — Silver fruit spoon * Mr. and Lady Georgiana Mure — Silver-mounted ink [sic] * Mrs. Gerald Fitzgerald — Silver-mounted inkstand * Mrs. Ruthven — Set of silver knives in case * Mrs. Blair — Umbrella * Mrs. Willie Lawson — Hunting crop * —— Three driving whips * —— Tea tray * Mr. and Mrs. Ramsay — Umbrella * Mr. George Hunt — Silver flower bowl * Mr. and Mrs. Reginald Cookson — Silver biscuit box * Mr. Arthur and V. James — Silver two-handled cup and cover * Mr. Robbio Stubber — Pair of silver scent bottles * Mr. and Mrs. Geo. Baird — Silver bowl * Mr. and Mrs. Harrison Broadley — Pair of silver flower vases * Mrs. Grant—Silver flower-pot stand * Mrs. Villiers — Silver corkscrew * Capt. Spender Clay — Antique silver snuffbox * Mr. and Mrs. Weir — Silver bacon dish * Mr. Baird — Pair of silver candlesticks * Mr. Athol Hay — Silver sugar bowl * Capt. Ewing — Pair of silver fruit dishes * Mr. and Mrs. C. J. Phillips — Pair of silver baskets * Miss Esmé Duncan — Silver box * Mr. and Mrs. Ronald Paton — lvory paper knife * Dr. Freshfleld — Work case * Mrs. Arkwright — Silver-mounted blotter * Mr. and Mrs. Peake — Silver-mounted stationery case * Miss Goddard — Book * Mr. D. Baird — Silver inkstand * J. G. and Jane B. Hay — lnkpot, with silver watch top * Mr. and Mrs. Wadsworth Ritchie — Pair of silver dishes in case * Mr. and Mrs. Guy Fenwick — Set of twelve silver knives in case * Jane and Uncle Willie — Silver sugar basin in case * Mr. and Miss Millington Knowles — Set of four silver dessert spoons in ease * Herbert and Lady Beatrix Herbert — Silver flower dish * Mr. and Mrs. J. B. Thorneycroft — Four silver candlesticks * Mr. and Mrs. Russell? M [illegible, ink has spread] — Silver bowl [Col. 2c / Col. 3a] * Mr., Mrs., and the Misses Wm. Cooper — Fan * Miss Winearls — Silver-mounted scent bottle * Sir Ernest Cassel — Diamond and enamel brooch * Mr. John S. Cavendish — Gold pencil case * —— Diamond and sapphire bracelet * Miss Lottie Coats — Diamond and pearl brooch * Hon. T. Robarts — Diamond brooch * Mr. and Mrs. Chas. E. Hay — Enamel and pearl miniature holder * Evelyn Ward — Cornomandel [sic] box * Mr. and Mrs. Slade — China clock * Lieut.-Col. Jervoise — Fan * Mr. and Mrs. J. B. Fergusson—Set of four silver menu holders * Mr. Guy R. F. Dawson — Silver card case * Rev. E. V. and Mrs. Hodge — Silver dish * Mr. C. S. and Mrs. Newton — Silver waiter * Mrs. Metcalfe — Gold, turquoise, and ruby brooch * Lord and Lady Erne — Set of three gilt decorated liqueur decanters * Mr. and Mrs. Chas. Grant — Two silver-mounted spirit decanters * Mr. and Mrs. George Baird — Set of three cut-glass decanters * Mr. Peter Cookson—Pair of silver-mounted decanters * Mrs. Featherstonehaugh — China ornament * Aunt Mary — China coffee service in case * Mr. H. S. Sykes — Silver-mounted telegram form case * Capt. Meade — Pair of engraved claret jugs * Lord and Lady Binning — Silver-mounted claret jug * Mr. and Mrs. Baldock — Silver-mounted water jug, with inscription * Mrs. and the Misses Chaplin — Pair of gilt decorated vases * —— Silver-mounted claret jug * Kittie, Margie, Hestie, Walter, Phillip, and Millicent Tanner — Pair of silver peppers case * Mr. J. R. J. Logan — Silver-mounted claret jug * Miss Ethel Baird — Painted china box * Mrs. D. A. Neilson — Pair of female figures with Cupids * M. M. Phillips — Painted china miniature box * Lady Waldie Griffith — Stationery case * —— Painted two-fold screen * Miss Mabel Fitzgerald — Silver-mounted vase * Major Bouverie — Silver-mounted match holder * —— Enamelled inkstand and candlesticks to match * Mrs. Duncan — Stationery case and blotter * —— Silver-mounted stationery case * —— Tortoiseshell and silver-mounted paper-knife * Miss Mills — Dresden china vase, cover, and stand * —— Six Vols. of Ruskin's "Modern Painters" * Mrs. W. Baird — Leather bag * Miss Langridge — Four silver spoons * Miss Kirk and Miss Hemsley — Silver-mounted photo frame * Miss Nessie Hemsley — Silver-mounted photo frame * Captain and Mrs. St. Aubyn Loftus — Silver vase * Decima Walker Leigh — Pair of silver-mounted menu stands * Mrs. Charles Thomson — Mirror in silver frame * Miss Reese — Silver crumb scoop * —— Silver-mounted seal and case * Mary Abercorn Alexander and Gladys Hamilton — Silver inkstand * Mr. and Mrs. Cecil Chaplin — Silver pen, pencil, and knife in case * Miss Gwendoline Brassey — Silver-mounted ice pail * Mr. and Mrs. and Misses Clifford Chaplin — Pair of silver candlesticks * Mr. and Mrs. Magee — lvory paper knife * Misses Dorothy and Maude Pilcher — Scent bottle * Miss Ashton — Silver-mounted clock * Mrs. William Clarence and Miss Watson — Silver crumb scoop * Major and Mrs. Ed. Baird — Egg-boiler on silver stand * Mr. A. F. H. Fergusson — Pair of silver coffee pots * —— Table mirror * —— Pair of silver vases * Mrs. R. B. Mnir — Silver fox ornament * Mr. H. Brassey and Mr. H. R. Molynenx — Silver teapot * —— Pair of silver sauce boats * Mr. and Mrs. Heathcote — Silver cream jug * Misses Thompson — Silver photo frame * Mr. C. D. Rose — Pair of silver fruit dishes * Mr. T. Archibald Hope — Silver toast-rack * Mr. and Mrs. Robert Hunt — Pair of silver sauce boats * Major and Mrs. Candy — Pair of silver fruit baskets * Misses Trefusis — Silver-mounted owl mustard-pot * Mrs. Frank Chaplin — Silver photo frame * Major Vaughan Lee — Silver waiter * Major Byng — Pair of silver menu stands * Lady Wilton — Silver photo stand * Geoffrey and Sibyll Palmer — Scent bottle * Dr. Clement Godson — Silver salad cruet * Mr. Mackenzie — Silver cigar case * Mr. G. Colvin White — Set of four silver trays * Mr. Edgar Brassey — Silver pipe lighter * Miss Emily Dawson — Photo frame * Mrs. Gerald FitzGerald — Silver match-box holder * A. Barns — Silver waiter * Miss Palmer — Letter-clip and dish * Mr. and Mrs. Aubrey Coventry — Photo frame * —— Silver bowl three feet * Mr. and Mrs. Hornsby — Openwork silver basket * —— Antique silver box * Mr. and Mrs. H. R. Baird — Silver coffee-pot * —— Pair of silver salts * Mr. Hugh Wanemley — Silver-gilt match-box * Captain Gordon Wilson — Silver snuff-box * Mrs. Whitelaw — Silver mustard-pot * Mrs. Palmer — Silver spoon * Mr. Dudley Majoribanks — Silver bowl and cover * Mr. Wilfred F. Ricardo — Pair silver candlesticks * Indoor Servants at Knossington Grange and 8, Rutland Gate — Breakfast warmer and two silver entree dishes and covers * Outdoor Servants at Knossington Grange — Silver stationery case * Mr. Waterman (coachman) — Driving-whip * Mr. Alexander (coachman) and Mrs. Alexander — lnk-stand * Villagers of Knossington — Silver sugar bowl, sugar tongs, and cream ewer in case * Silver vase, with inscription — "Capt. Mann Thomson, Royal Horse Guards, from the Estate and Household at Dalkeith, on the occasion of his marriage, 25th July, 1901." * Miss Baldock — Pair of scent bottles * Captain Cook — Paper-knife * Sir A. Baird — Pair of silver muffineers * Rev. H. W. Trower — Pair of silver peppers * Mr. T. Vandeleur — Silver cigarette box * Lady Miller — Silver milk jug * Mr. Hedworth Barclay — Silver muffineer * Miss May A. Jackson — Photo frame * Mr. Geoffrey Heneage — Silver ash tray * Mr. and Mrs. R. B. Hay — Pair silver mustard-pots * Mrs. George Charteris — Silver-mounted calendar * Royal School of Art Needlework, Exhibition-road — Silvered copper heart-shaped box * Mr. A. C. Newbigging — Silver fox ornament * Mr. S. Schreiber — Silver match box * Mr. and Mrs. J. H. J. Phillips — Silver muffineers * Mr. and Mrs. Fyfe Jameson — Silver flask * Mrs. Beaumont Lubbock — Silver bon-bon dish * Lord Castlereagh — Salad bowl * Captain Hambro — Silver card case * Lord Longford — Silver bowl * Captain —— Silver waiter * Mrs. Forester — Silver frame * Mrs. Martin — Tea cloth * Mr. and Mrs. Cooper — Whip * Earl Lonsdale — Silver tray * Lady Augusta Fane — Red box * Mr. Paul Phipps — Clippers * Mr. E. Herlick — lnkstand<ref>"Marriage of Captain Mann Thomson and Miss Duncan." ''Grantham Journal'' 27 July 1901 Saturday: 2 [of 8], Cols. 2a–3b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000400/19010727/003/0002.</ref> </blockquote> ===August 1901=== ====30 August 1901, Friday==== [[Social Victorians/People/Horos|The Horoses]] (troublesome members of the Golden Dawn) were thrown out of 99 Gower Street and moved to Gloucester Crescent (King 89 91). ===October 1901=== ====31 October 1901, Thursday==== Halloween. ===November 1901=== ====5 November 1901, Tuesday==== Guy Fawkes Day ===December 1901=== ====25 December 1901, Wednesday==== Christmas Day ====26 December 1901, Thursday==== Boxing Day ===Works Cited=== *[1901-02-23 Cheshire Observer] "Duke of Westminster. Brilliant Function." Cheshire Observer 23 February 2901, Saturday: 6 [of 8], Col. 1a–6c [of 8]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000157/19010223/114/0006 (accessed July 2019). *[1901-04-25 Stage] "Provinces." "Amateurs." The Stage 25 April 1901, Thursday: 11 [of 24], Col. 3c, 4b–c [of 5]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001179/19010425/028/0011 (accessed July 2019). *[1901-07-24 Beverley Echo] "Stray Notes." Beverley Echo 24 July 1901, Wednesday: 2 [of 4], Col. 4b [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001561/19010724/037/0002 (accessed July 2019). ==1902== Sometime in 1902, London publisher [[Social Victorians/People/Working in Publishing#George Newnes|George Newnes]] published an edition of [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s first (1892) collection of Holmes stories. ===January 1902=== ====1 January 1902, Wednesday, New Year's Day==== ===February 1902=== ==== 13 February 1902, Thursday ==== King Edward VII and Queen Alexandra were present with some of their friends at Niagara, which must have been an ice-skating rink. Mr. and [[Social Victorians/People/Churchill|Mrs. George West]] are Lady Randolph Churchill and George Cornwallis-West.<blockquote>SOCIAL & PERSONAL Royalty at Niagara. Quite a record audience was present at Niagara yesterday, when the free skating and waltzing competitions were skated off to the sound of gay music in a brightly lighted, warm atmosphere. The royal box made a goodly show with its trappings of Oriental hangings and decorations of palms. The Royal Box. The King and Queen were accompanied by Princess Victoria and Prince and Princess Charles of Denmark, the Prince and Princess of Wales having previously arrived. Their Majesties were conducted to the spacious box by Mr. Hayes Fisher. All the royal ladies wore black, the Queen adding a bunch of yellow Lent lilies to her sombre attire. Her two daughters lightened their mourning with touches of white, and the Princess of Wales wore a bunch of violets in her toque, with a twist of white. In the adjoining box, among members of the suite were the Countess of Gosford, Earl Howe, Mr. Sidney Greville, Mr. H. J. Stonor, Lieut.-Colonel Davidson, Lieut.-Colonel Legge, and Viscount Crichton. In boxes on the other side of the royal box were Lady Alice Stanley, with the Ladies Acheson, the Countess of Derby, Countess De Grey and Lady Juliet Lowther, [Col. 3c/4b] Mr. and [[Social Victorians/People/Churchill|Mrs. George West]] [Lady Randolph Churchill and George Cornwallis-West], Sir Edgar and Lady Helen Vincent, the Duchess of Bedford and the Marquis of Tavistock, [[Social Victorians/People/de Soveral|M. de Soveral, the Portuguese Minister]], and Viscount and Viscountess Falmouth. Others to be picked out in the crowd were Consuelo Duchess of Manchester, Viscountess Coke and Mrs. Ellis, Lady Archibald Campbell and her son, Mrs. Grenander, Lord and Lady Lilford, Mr. and Mrs. Edward Stonor, Mrs. [[Social Victorians/People/Bourke|Algernon Bourke]], Mr. Algernon Grosvenor, and Mr. and Mrs. Hwfa Williams. The royal party took a great interest in the contests, and especially applauded the Swedish couple in their graceful evolutions. Their Majesties remained over an hour, the royal party taking their departure shortly after five.<ref>"Social & Personal." ''Daily Express'' 14 February 1902, Friday: 4 [of 8], Cols. 3c–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020214/088/0004. Print p. 4.</ref></blockquote> ===March 1902=== The last time Bret Harte and Arthur Collins saw each other: "They dined at the Royal Thames Yacht Club, and Collins found his 'poor old friend' 'saldly aged and broken, but genial and kind as ever.' They sat an hour at a music hall and Harte wrote afterwards to thank Collins for having 'forced him out.'" (Nissen, Axel. Bret Harte: Prince and Pauper. Jackson, MS: U P of Mississippi, 2000: 262) ===April 1902=== ====9 April 1902, Wednesday==== According to a letter to Lady Gregory, [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] dictated "2000 words in an hour and a half" "to a typewriter; he was working on his novel (Wade 370). At this point, a typewriter was a person who used the machine called typewriter to type. ====10 April 1902, Thursday==== [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote to Lady Gregory from 18 Woburn Buildings about working on his novel "-- dictating to a typewriter" (Wade 370). ===May 1902=== ====5 May 1902, Monday==== Bret Harte died. Arthur Collins does not seem to have been there at his death; “his dear friend Madame Van de Velde and her attendants” were, though (Pemberton, T. Edgar. The Life of Bret Harte. Dodd, Meade, 1903. http://books.google.com/books?id=eZMOAAAAMAAJ). Not sure when the funeral occurred, but he is buried “in quiet Frimly churchyard,” (341) and <quote>In accordance with his well-known views on such subjects the funeral was a very simple one. Among the few who followed him to his ivy-lined grave were Mrs. Bret Harte, his son and daughter-in-law, Mr. and Mrs. Francis King Harte, his daughter, Miss Ethel Harte, Madame Van de Velde, Colonel Collins, Mr. A.S. Boyd, and a small cluster of grief-stricken friends.</quote> (Pemberton, T. Edgar. The Life of Bret Harte. Dodd, Meade, 1903. http://books.google.com/books?id=eZMOAAAAMAAJ (accessed November 2014). ====8 May 1902, Thursday==== <quote>On Thursday, May 8, 1902, in the squat, mid-Victorian church of St. Peter's in the Surrey village of Frimley, a group of about twenty people had come to show their final respects to Francis Bret Harte. Outside it was raining steadily . In the subdued light from the stained-glass windows, one cold discern a small group at the front of the church consisting of Anna Harte, her son Frank, her daughter-in-law Aline, and her daughter Ethel. Another small group was formed around Madame Van de Velde, including one of her unmarried daughters, Miss Norris (the sister of her son-in-law Richard Norris), and Mrs. Clavering Lyne. Of Harte's closest friend, only Arthur Collins and Alexander Stuart Boyd were present. Pemberton had written to Frank the day before that he wished to attend the funeral but that in his "deplorable state of health" it was impossible for him to travel. Beside the small group of family and old friends, the rest of the people who heard the service conducted by the rector of Frimley, Reverend W. Basset, were recent acquaintances from among the local gentry. As one newspaper noted: "The funeral was of the simplest possible character and the phrase 'this our brother' had a peculiar poignancy, for, though a group of villagers stood in the rain under the trees as the hearse arrived, there were few in the church, who had not the right to call Mr. Bret Harte friend." The simplicity of the service was in keeping with Bret Harte's wishes.</quote> (Nissen, Axel. Bret Harte: Prince and Pauper. Jackson, MS: U P of Mississippi, 2000: 263) ===June 1902=== Summer 1902: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until Yeats bought the Tower at Ballylee. (I got this from Wade?) ====3 June 1902, Tuesday==== [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote Arnold Dolmetsch, asking him to "chair ... a lecture he [was] soon to give": "You are the only one, I suppose, in the world now, who knows anything about the old music that was half speech, and I need hardly say that neither [[Social Victorians/People/Florence Farr|Miss Farr]] nor myself, could have done anything in this matter of speaking to notes without your help" (Campbell 142). ====7-9 June 1902, Saturday-Monday==== The [[Social Victorians/People/Warwick|Earl and Countess of Warwick]] hosted a house party: <quote>The Earl and Countess of Warwick entertained a distinguished house party from Saturday to yesterday, including the Grand Duke Michael of Russia and the Countess of Torby, the Earl and Countess of Craven, the Earl and Countess of Kilmorey, Earl Cairns, Lord and Lady Savile, Lord Chesham, Sir Frederick and Lady Milner, Colonel and Lady Gwendoline Colvin. Lady Margaret Orr-Ewing, Lady Eva Dugdale. Mrs. Kenneth Wilson, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Right Hon. H. Chaplin, M.P., Hon. H. Stonor, Mr. J. Pease, M.P., Captain Brinton, and Captain J. Forbes.</quote> (1902-06-10 Manchester Courier and Lancashire General Advertiser) ====10 June 1902, Tuesday==== [[Social Victorians/People/Florence Farr|Florence Farr]]'s first public performance in which she "recit[ed] to her own accompaniment on the psaltery was at the Hall of Clifford's Inn, Fleet Street, on 10 June 1902 (Campbell 144, n. 18). ==== 12 June 1902, Thursday==== 12 June 1902: <quote>Thursday, the 12th inst., being the grand day of Trinity term at Gray's-inn, the Treasurer (Mr. Herbert Reed, K.C.) and the Masters of the Bench entertained at dinner the following guests: The Right Hon. Lord Strathoona and Mount Royal, the Right Hon. Lord Avebury, the Right Hon. H. H. Asquith, K.C, M.P., the Right Hon. Sir Frank Lascelles, G.C.B. (British Minister at Berlin), General Sir Edward Brabant, K.C.B., the Right Hon. Sir Edward Carson (Solicitor-General), Sir Squire Bancroft, Colonel Alfred Egerton, C.B. (Equerry to H.R.H. the Duke of Connaught), Mr. Austen Chamberlain,M.P., Colonel Royds, M.P., and Mr. Frank Dicksee, R.A. The Benchers present in addition to the Treasurer were H.R H. the Duke of Connaught, Lord Ashbourne, Lord Shand, Mr. Henry Griffith, Sir Arthur Collins, K.C, Mr. Hugh Shield, K.C, His Honour Judge Bowen Rowlands, K.C, Mr. James Sheil, Mr. Arthur Beetham, Mr. John Rose, Mr. Paterson, Mr. Mulligan, K.C, Mr. Mattinson, K.C, Mr. Macaskie, K.C., Mr. C. A. Russell, K.C., Mr. Montague Lush, K.C., Mr. Dicey, C B., Mr. Barnard, Mr. H. C. Richards, K.C., M.P., Mr. Duke, K.C., M.P., Sir Julian Salomons, K.C., with the Preacher (the Rev. Canon C. J. Thompson, D.D.).</quote> (The Solicitor's Journal and Reporter. June 21, 1902. Volume XLVI. 1901-1902 [November 2, 1901, to October 25, 1902]: 588. Google Books: http://books.google.com/books?id=9T84AQAAIAAJ&pg=PA588). ====26 June 1902, Thursday==== Edward VII crowned King of England. 26 June 1902. There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in May 1902, so the celebration with Harte present did not take place this year. Did it take place at all? ===July 1902=== ====3 July 1902, Thursday==== [[Social Victorians/People/Mathers|MacGregor and Moina Mathers]] were living at 28 Rue Saint Vincent, Buttes Montmartre, Paris (Howe 244). ===September 1902=== ''Tristan and Isolde'' at the Covent Garden. ====25 September 1902, Thursday==== "There were no winter performances of opera at Covent Garden in those times .... In 1902 an autumnal series was added, and there were several Wagner nights, the last of which was on Thursday, 25 September, when Philip Brozel and Blanch Marchesi were starred in ''Tristan and Isolda'' with Marie Alexander as Brangane" (Baring-Gould II 704, n. 14, quoting Rolfe Boswell). ===October 1902=== ==== 24 October 1902, Friday ==== The ''Daily Express'' reported on the annual opening of the Prince's ice-skating rink, revealing who had an interest in skating:<blockquote>PRINCE’S RINK OPENS. The first ice of the season was skated upon yesterday. It was the carefully-prepared ice which Mr. H. W. Page and Mr. Nightingale offer to the members of Prince’s Skating Club, in Knightsbridge, and was in grand condition. The [[Social Victorians/People/Bourke|Hon. Algernon Bourke]] opened the rink for the seventh season, and in the afternoon and evening the West End patronized the popular club to skate or to lounge to the pleasant strains of the Viennese band. [[Social Victorians/People/Princess Louise|Princess Louise]] is again at the head of the ladies’ committee, with the [[Social Victorians/People/Portland|Duchess of Portland]] and [[Social Victorians/People/Londonderry|Marchioness of Londonderry]] as co-members, and Lord Edward Cecil and many other well-known skaters are identified with the committee work. The skating hours are from 9.30 to 1 and 3 to 7, and on Sundays 3 to 7 only.<ref>"Prince's Rink Opens." ''Daily Express'' 25 October 1902, Saturday: 5 [of 8], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19021025/132/0005.</ref></blockquote> ====31 October 1902, Friday==== Halloween. ===November 1902=== ====5 November 1902, Wednesday==== Guy Fawkes Day. ==== 8 November 1902, Saturday ==== The Earl and Countess of Warwick hosted a shooting party at Easton Lodge:<blockquote>The [[Social Victorians/People/Warwick|Earl and Countess of Warwick]] are entertaining a large party at Easton Lodge this week-end for [?] shooting, and among their guests are the Grand Duke Michael of Russia and Countess Torby, the Duc d'Alba, the Duke of Sutherland, Earl Howe, Earl Cairns, Lord Dalmeny, Lord Herbert Vane-Tempest, the Hon. John and Lady [Choely?] Scott-Montagu, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Right Hon. Henry Chaplin, M.P., General and Mrs. Arthur Paget, and Miss Leila Paget, Miss Naylor, Miss Deacon, and Mr W. M. Low.<ref>"Guests at Easton Lodge." ''Birmingham Mail'' 08 November 1902, Saturday: 2 [of 6], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000644/19021108/091/0002. Print title: ''Birmingham Daily Mail''; p. 2.</ref></blockquote> ====29 November 1902, Saturday==== [[Social Victorians/People/Muriel Wilson|Muriel Wilson]]’s cousin, Lady Hartopp, was involved in a divorce case: <blockquote>Society Women in a Law Court Case. Mr. Justice Barnes’s Court is now crowded by society people. What is the strange fascination which brings elegantly dressed ladies, accustomed to luxurious surroundings and all the external refinements of life, to sit for hours in stuffy court, where the accommodation is all the plainest, and the surroundings are none too attractive. It would need some assurance to invite a Belgravian Countess, or the wife of Mayfair Millionaire to spend the morning under such conditions unless there were the attraction of a very strong piece of scandal. One could not presume to suggest she should attend Missionary meeting, or social reform movement, under any such conditions. At least I must confess that I never heard of one being packed with a West End crowd as the Court just now. Of course it cannot be mere idle curiosity. Our higher education for girls must have cured Mother Eve’s failing long ago. Cynics suggest that it is the survival in our highly-civilised modern conditions of that instinct of the wild creature which incites attack on the wounded or injured fellow. Wild birds will sometimes peck injured bird to death. Are these fair and soft-voiced ladies animated by the same spirit when they throng witness the ordeal through which a woman of their own class is passing? The Latest Divorce Case. Lady Hartopp, the heroine of the story which has been occupying the tongues and thoughts of the upper ten thousand for the last 48 hours, is a member of a well-known and wealthy family, and is herself remarkable for her beauty. Her two sisters are as famous for their charms as herself, and society has given them many flattering titles. The daughters of Mr. C. H. Wilson, the great shipowner, whose sails are on every sea, are as favoured by Fortune as Venus. Miss Muriel Wilson, the society beauty, is a cousin of Lady Hartopp, and Lady Chesterfield is her sister. It was at Tranby Croft, near Hull, the residence of Mr. and Mrs. Arthur Wilson, that the famous baccarat case occurred some years ago. Lady Hartopp is the niece of Mr. Arthur Wilson, and no doubt recollects that incident, and all the consequent stir. It attracted all the more notice at the time, because the then Prince of Wales had taken part in the game; but the Prince, who had nothing to be ashamed of, with characteristic straightforwardness, asked to go into the box and state all he knew. (1902-11-29 Norwich Mercury)</blockquote> ===December 1902=== ====16 December 1902, Tuesday==== A poem satirizing Florence Farr and Arnold Dolmetsch was published in ''Punch''. ====25 December 1902, Thursday==== Christmas Day ====26 December 1902, Friday==== Boxing Day ===Works Cited=== *[1902-06-10 Manchester Courier and Lancashire General Advertiser] "Court and Personal." Manchester Courier and Lancashire General Advertiser 10 June 1902, Tuesday: 5 [of 10], Col. 3c [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000206/19020610/033/0006 (accessed July 2019). *[1902-11-29 Norwich Mercury] "Society Women in a Law Court Case." And "The Latest Divorce Case." Norwich Mercury 29 November 1902, Saturday: 5 [of 12], Col. 1b [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001669/19021129/072/0005 (accessed July 2019). ==1903== From sometime in 1891 to sometime in 1903 Eduoard de Reszke was "a leading bass" at the New York Metropolitan Opera (Baring-Gould II 112, n. 114). "[I]n England in 1903, gramophone distinctly meant the Berliner-Gramophon & Typewriter disc machine, while cyclinder machines were known as phonographs or graphophones " (Baring-Gould II 745, n. 15). Gerald Balfour was "largely responsible for getting the important Land Acts of 1903 under way" (O'Connor 163). ===January 1903=== ====1 January 1903, Thursday, New Year's Day==== ====3 January 1903, Saturday==== Madame Troncey was doing a portrait of [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] (Wade 392). ===June 1903=== Summer 1903: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). === August–September 1903 === ==== 20 and 25 August and 3 September 1903 ==== The 1903 America's Cup yacht race in New York Harbor with Nathaniel Herreshoff's ''Reliance'' for the US and Sir Thomas Lipton's ''Shamrock III'' for the UK,<ref>{{Cite journal|date=2022-09-11|title=1903 America's Cup|url=https://en.wikipedia.org/w/index.php?title=1903_America%27s_Cup&oldid=1109663279|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/1903_America%27s_Cup.</ref> the 12th challenge for the cup and "the most expensive Cup challenge in history."<ref name=":0">{{Cite web|url=https://www.americascup.com/history/26_LIPTONS-THIRD-CHALLENGE|title=LIPTON’S THIRD CHALLENGE|last=Cup|first=America's|website=37th America's Cup|language=en|access-date=2024-07-02}} https://www.americascup.com/history/26_LIPTONS-THIRD-CHALLENGE.</ref> The first race was run on 20 August 1903, the 2nd on 25 August and the 3rd on 3 September.<ref name=":0" /> Because the ''Reliance'' won the first 3 races, the best 3-out-of-5 race ended after the 3rd one. ===October 1903=== Sometime in October 1903, [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s "The Adventure of the Empty House," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 331). ====31 October 1903, Saturday==== Halloween. ===November 1903=== Sometime in November 1903 Arthur Conan Doyle's "The Adventure of the Norwood Builder," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 415). ====5 November 1903, Thursday==== Guy Fawkes Day ===December 1903=== Sometime in December 1903 Arthur Conan Doyle's "The Adventure of the Dancing Men," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 529). ====16 December 1903, Wednesday==== "On 16 December, Punch satirised an activity in which Dolmetsch was concerned. [[Social Victorians/People/Florence Farr|Florence Farr]] was acting as secretary for a newly-formed fellowship known as 'The Dancers', a body whose aim was to 'fight the high and powerful devil, solemnity'. In a poem entitled L'Allegro up to date, the final stanza is devoted to Dolmetsch: <poem>:The old forgotten dancing-lore, :The steps we cannot understand, :DOLMETSCH agrees to take in hand, :These on the well-trod stage anon, :When next our learned sock is on, :We’ll show, while ARNOLD, Fancy’s child, :Tootles his native wood-wind wild.</poem> This verse is curiously prophetic for Dolmetsch had not yet introduced the recorder into his concerts, although he occasionally included a flute. Dolmetsch did know something of the steps of the old dances but it was his wife who later researched the subject most thoroughly and wrote two books on the subject." (Campbell 151–52) ===25 December 1903, Friday=== Christmas Day ====26 December 1903, Saturday==== Boxing Day ===Works Cited=== *Baring-Gould. *Campbell. ==1904== ===January 1904=== Sometime in January 1904 [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s "The Adventure of the Solitary Cyclist," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 399). ===March 1904=== Sometime in March 1904 Arthur Conan Doyle's "The Adventure of Black Peter," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 384). ===April 1904=== Sometime in April 1904, Arthur Conan Doyle's "The Adventure of Charles Augustus Milverton," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 558, n. 1, and 559). ===June 1904=== Sometime in June 1904 Arthur Conan Doyle's "The Adventure of the Three Students," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 370). Summer 1904: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ===July 1904=== Sometime in July 1904, Arthur Conan Doyle's "The Adventure of the Golden Pince-Nez," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 351). ===August 1904=== Sometime in August 1904, Arthur Conan Doyle's "The Adventure of the Missing Three-Quarter," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 476). ===September 1904=== Sometime in September 1904, Arthur Conan Doyle's "The Adventure of the Abbey Grange," illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 491). ==1905== ===April 1905=== ====3 April 1905, Monday==== [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote to Lady Gregory from Dublin, saying he had "dictated a rough draft of a new Grania second act to Moore's typewriter" (Wade 368). ===June 1905=== Summer 1905: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ===July 1905=== ====10 July 1905, Monday==== 1905 July 10, the Austro-Hungarian Ambassador hosted a dinner party:<blockquote>The Austro-Hungarian Ambassador entertained the Duke and Duchess of Connaught and Princess Patricia of Connaught at dinner at the Embassy in Belgrave-square on Monday evening. There were also present the Spanish Ambassador and Mme. Bernabé, the United States Ambassador and Mrs. and Miss Whitelaw Reid, Princess Hohenlohe, Prince Francis of Teck, Princess Teano, the Earl of Essex, the Earl and Countess of Crewe, Viscount Villiers, Viscount Errington, Viscount Newry, Mrs. J. Leslie, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. R. Graham, Mrs. Astor, Lady Maud Warrender, Prince Furstenburg, Count Szenchenyi, Captain A. Meade, and Miss Pelly and Colonel Murray in attendance on the Duke and Duchess.<ref>"Court Circular." ''Times'', 12 July 1905, p. 7. ''The Times Digital Archive'', http://tinyurl.galegroup.com/tinyurl/AHRNq6. Accessed 20 June 2019.</ref></blockquote> ===October 1905=== ==== 1905 October 14, Saturday ==== A "send-off dinner" for Jerome K. Jerome before his trip to the U.S. occurred at the Garrick Club "the other evening" before October 14:<blockquote>Jerome K. Jerome has undertaken a six months lecturing tour in the United States. I believe that this tour will be a great success, particularly when the Americans come to realise that Mr. Jerome is not only a humorous writer but a brilliant, serious writer with very genuine pathos. His appeal on this side has not, perhaps, gone home to the English people as much as it should, but the quick-witted Americans will not be slow to recognise his talents of both kinds, nor will they fail to appreciate the significance of the fact that the other evening a send-off dinner was given to Mr. Jerome at the Garrick Club. The hosts of the evening were Mr. Pett Ridge and Mr. W. W. Jacobs, which shows that there is no such thing as literary jealousy among our best humorists. The presence of quite a galaxy of novelists to the dinner to Mr. Jerome, including Mr. Barrie, Sir Arthur Conan Doyle, Mr. Max Pemberton, Mr. H. G. Wells, Mr. G. B. Burgin, Mr. Arthur Morrison, and Mr. Israel Zangwill, serve to indicate the existence of a pleasant brotherhood among the writers of fiction. The readers of ''Three Men in a Boat'' may be interested to know that there were also present Mr. Jerome's companions in that famous journey — Mr. Carl Hentschel and Mr. C. Wingrove. When I have named further the presence of three artists in Mr. A. S. Boyd, Mr. John Hassall, and Mr. Will Owen, and two journalists in Dr. Robertson Nicoll and [[Social Victorians/People/Rook|Mr. Clarence Rook]], I have given some record of an exceedingly pleasant dinner party. The essential point, however, of this enumeration of names is that many of them are among the most highly honoured of Englishmen in the United States, and that thus Mr. Jerome cannot fail to reap additional benefit from this dinner so thoughtfully given in his honour by Mr. Jacobs and Mr. Pett Ridge.<ref>S., C. K. "A Literary Letter." ''The Sphere'' 14 October 1905, Saturday: 16 [of 20], Col. 2a–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001861/19051014/022/0016. Print p. 56.</ref></blockquote> ===November 1905=== Sometime in November 1905, "Arnold Dolmetsch was again asked to provide music for a Been Greet season in New York — an engagement that brought about his first meeting with two young actors on their first American tour, Sybil Thorndike, and her brother, Russell" (Campbell 169). Dolmetsch's return to the US; was [[Social Victorians/People/Horniman|Annie Horniman]] still with the Thorndikes? ==1906== ===March 1906=== ====5 March 1906==== "Mr. Frederick John Horniman, who died on March 5, in his seventy-first year, was the son of that well-known Quaker and tea-merchant, John Horniman, who made a magnificent fortune by retailing tea in air-tight packets, and, like his father, devoted both time and wealth to charitable objects. A great traveller, both for business and pleasure, Mr. Horniman gathered togther an admirable collection of curios, and this is housed at Forest Hill in the museum that bears his name. His private benefactions were also large. Mr. Horniman, who was a Liberal, sat in two Parliaments, representing Penrhyn and Falmouth Boroughs in one. He did not seek re-election in January last." ("The World's News." Illustrated London News (London, England), Saturday, March 10, 1906; pg. 338; Issue 3490, Col. C) ===June 1906=== Summer 1906: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ==1907== ===April 1907=== April 1907, [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] went to Italy with Lady Gregory (Harper 80 28). ===June 1907=== Summer 1907: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). '''1907 June 22, Saturday''' The annual dinner of the Correctors of the Press was held at De Keyser's Royal Hotel:<blockquote>CORRECTORS OF THE PRESS. The London Association of Correctors of the Press held their annual dinner at De Keyser’s Royal Hotel on Saturday. The Chairman was the Lord Mayor, and among his supporters were Sir John Cockburn, Colonel David Bruce, Colonel Earl Church, Lieutenant-Colonel Alsager Pollock, Sheriff Dunn, Mr. J. W. Cleland, M.P., Mr. R. Donald, Mr. T. Seccombe, Mr. Francis H. Skrine, Major H. F. Trippel, Mr. Walter Haddon, Mr. W. Pett Ridge, Mr. W. H. Helm, Mr. R. Warwick Bond, Mr. F. W. Rudler, Major Vane Stow, [[Social Victorians/People/Rook|Mr. Clarence Rook]], Mr. J. Randall (Chairman of the Association), Mr. Foxen, and Mr. Feldwick. Proposing the toast of "Literature,” Mr. W. H. Helm speculated as to what would follow the banning of "Mary Barton" by the Education Committee of the London County Council. In his opinion "The Swiss Family Robinson" was a more immoral book, because beyond any other work it had fostered the Micawber view of life. (Laughter.) The LORD MAYOR [init caps large, rest sm, throughout], submitting the toast of "The Readers' Pension Fund,” apologised for appearing in morning dress. The reason was that he had been to the King’s Garden Party at Windsor, and whlle he was returning to London by motor something burst. (Laughter.) Only that morning he had arrived from Berlin, where he learned some lessons useful to people who give dinners. When the Oberburgomeister of Berlin proposed the health of, say, the Lord Mayor of London, there was an end of the business. He did not push forward the Houses of Parliament, the Navy and Army, or even Literature. (Laughter.) Being a practical people the Germans when they met for a particular purpose applied themselves to no other, and the English would well to copy them. (Hear, hear.) Mr. J. RANDALL said that last year the Association helped five readers and one reader’s widow to pensions, and this year it had done the same for two readers and two widows. One of the men assisted last March had taught himself Greek, Arabic, and Sanscrit, and in leisure moments amused himself by making object glasses for microscopes and telescopes. At this very gathering there was a printer’s reader who was Hebrew scholar. (Hear, hear.) With regard to finance Mr. Randall was happy to say that this dinner would enable the Association to establish a fourth pension. (Cheers.) The Lord Mayor, [[Social Victorians/People/Borthwick|Lord Glenesk]] (President of the Readers' Pensions Committee), the Clothworkers’ Company, and the Cutlers’ Company had contributed ten guineas each, and the total addition to the fund resulting from the dinner was £l90. During the evening excellent entertainment was provided by Miss Helena Foxen, Miss Kathleen Dwyer, Mr. T. C. Bell, Mr. P. E. Syrett, Mr. Prank Rhodes, and Mr. E. Croft-Williams, the last-named being the hon. musical director.<ref>"Correctors of the Press." ''Morning Post'' 24 June 1907, Monday: 4 [of 14], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19070624/074/0004. Print p. 4.</ref></blockquote> ===November 1907=== ====10 November 1907==== <quote>On 10 November, Dolmetsch, 'awfully tired and disquieted with overwork', writes to Horne, 'longing for Florence'. 7, Bayley Street<br />W.C.<br />My concert went very well last night. Melodie quite distinguished herself, and a sister of [[Social Victorians/People/George Bernard Shaw|Bernard Shaw]] Lucy Carr Shaw sang delightfully. …<br />But Symmons [sic] … did not go before 1 o'cl. and yet, by the first post this morning, I got a charming poem on Rameau. … He must have spent all night on it.</quote> (Campbell 120) ==1908== In 1908 Sidney Paget died in 1908 in some "untimely" fashion (Baring-Gould II 239). === April 1908 === ==== 1908 April 9, Thursday ==== The Provisional Committee for the Shakespeare Memorial demonstration at the Lyceum Theatre met at the Hôtel Métropole:<blockquote>SHAKESPEARE MEMORIAL. A meeting of the Provisional Committee for the forthcoming Shakespeare Memorial demonstration at the Lyceum Theatre was held yesterday at the Hôtel Métropole. Mr. T. P. O’Connor, M.P., presided, and there were present : The Earl of Lytton, Mr. Percy Alden, M.P., Mr. Henry Ainley, Mr. Percy Ames, Mr. Robert Barr, Mr. Arthur à Beckett, Mr. Austin Brereton, Mr. Acton Bond (General Director of the British Empire Shakespeare Society), Mr. Dion Boucicault, Mrs. Bateman-Crowe, Professor Boss, Mr. Norreys Connell, Mr. W. M. Crook, Mr. John Cutler, K.C., Mr. J. Comyns Carr, Mr. Ernest Carpenter, the Rev. P. H. Ditchfleld, Mr. Robert Donald, Mr. A. C. Forster Boulton, M.P., Mr. and Mrs. Laurence Gomme, Mr. A. A. Gardiner, Mr. C. T. Hunt (hon. secretary London Shakespeare League), Mr. Laurence Housman, Mr. J. A. Hobson. Mr. Ford Madox Hueffer, Mr. Selwyn Image, Mr. Henry Arthur Jones, Mr. Jerome K. Jerome, Mr. Frederick Kerr, Miss Gertrude Kingston, Professor Knight, Mr. Matheson Lang, the Hon. Mrs. Alfred Lyttelton, Miss Lillah McCarthy, Mr. Justin Huntly McCarthy, Colonel Henry Mapleson, Dr. Gilbert Murray, Mr. T. Fairman Ordish, Mr. A. W. Pinero, Mr. Ernest Rhys, [[Social Victorians/People/Rook|Mr. Clarence Rook]], the Rev. J. Cartmel Robinson, Mr. George Radford, M.P., Mr. Clement Shorter, Mr. Otto Salimann (hon. secretary of the Elizabethan Society), [[Social Victorians/People/George Bernard Shaw|Mr. Bernard Shaw]], Mr. H. W. Smith, Mr. Herbert Trench, [[Social Victorians/People/Todhunter|Dr. Todhunter]], and Mr. James Welch. It was agreed that the Lyceum demonstration should take place in May, and a resolution should be moved in favour of the establishment of a National Theatre as a memorial to Shakespeare.<ref>"Shakespearea Memorial." ''Morning Post'' 10 April 1908, Friday: 7 [of 12], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19080410/126/0007. Print p. 7.</ref></blockquote> ===June 1908=== Summer 1908: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). ==== 1908 June 21, Sunday ==== Very large demonstration for women's suffrage in Hyde Park coming from "seven points in London."<blockquote>WOMAN'S VOTE. SUFFRAGISTS' GREAT MARCH TO HYDE PARK TODAY. WHITE DEMONSTRATION. AMUSING ADDRESS TO M.P.'s. FROM RIVER LAUNCH. From seven points in London to-day seven big prossesions will march to Park, and there jointly demand the Parliamentary franchise for women. The whole town will be alive with demonstrating suffragists. The streets will resound with the cry of "Votes for Women." In Hyde Park eighty speakers will voice the demand from twenty platforms. London has been divided into districts for the purposes of the mighty demonstration, and each of theee has an assembling place, from which the processions will move off to Hyde Park, as given in the following official list: — A. — Euston-road. — Form up at 12 o'clock, east of Euston Station. Start at 1 p.m. March via Euston-road, Portland-place, Upper Regent-street, Oxford-street, to the Marble Arch. B. — Trafalgar-square. — Form up 12.30. Start 1.30. March via Pall Mall, Regent-street, Piccadilly, Berkeley-street, and Mount-street to the Grosvenor Gate. C. — Victoria Embankment. [sic] Form up 12.30. Start from Westminster Bridge 1.30. March via Victorla-street, Grosvenor-place, to Hyde Park Corner. D. — Chelsea Embankment. — Form up 12.30. Start 1.30. March via Oakley-street, King's-road, Sloane-square, Sloane-street to Albert Gate. E. — Kensington High-street. — Form up 1 o'clock. Start 1.30. March via Kensington into the Alexandra Gate of the Park. F. — Paddington Station. — Form up 1 p.m. Start 2 p.m. March via Victoria Gate into Hyde Park. G. — Marylebone-road. — Form up 12.30. Start 1.30. March via Seymour-place, Seymour-street, and Into the Park close to the Marble Arch. The demonstrators will come from all parts of the country, some seventy special trains being run from the big towns in the provinces. These will be met at the London stations by white-garbed "Captains" and "Stewards," and their occupants marshalled in proper divisions. Literature and the drama will be represented in several of the processions. Mr. and Mrs. Bernard Shaw will join in Trafalgar-square, and so will Mr. Pett Ridge. Starting from Euston-road will be a coach carrying Mrs. Parkhurst, Miss Beatrice Harraden, Mrs. Mona Caird, and Miss Elizabeth Robins. Mrs. Israel Zangwill will chaperon a party on a coach from the Thames Embankment, which will include Professor and Mrs. Ayrton, Madame Sarah Grand, Miss Lillah McCarthy (Mrs. Granville Barker), Miss Marian McCarthy, Mr. Lucien Wolf, Professor Perry, F.R.S. (scientist), Mrs. H. G. Wells, Mrs. Alice Meynell, and Suffragist leaders from Sweden, Finland, and Norway. In Finland women not only have the vote, but they sit in Parliament. Madame Stromberg, from that country, is now in London attending the Horse Show at Olympia, and will be present at to-day's demonstration. Mr. H. Nevinson and Mr. H. N. Brailsford will walk in the Embankment procession. On the Kensington four-in-hand coach will be:— [[Social Victorians/People/Rook|Mrs. Clarence Rook]], Mrs. Jopling Rowe, Mlle. Stavance (Norwegian editor and authoress), Mrs. French Sheldon, F.R.G.S., and Miss Christine Silver. ... In addition to seven four-horse coaches — one for each procession — there will be more than sixty brakes, filled with country suffragists, and elaborately decorated. [Story continues.]<ref>"Women's Vote. Suffragists' Great March to Hyde Park To-day. White Demonstration. Amusing Address to M.P.'s from River Launch." ''Lloyd's Weekly Newspaper'' 21 June 1908, Sunday: 1 [of 28], Col. 1a–c [of 5], 2, Col. 5. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003216/19080621/002/0001. Print p. 1.</ref></blockquote> ===Works Cited=== *Baring-Gould ==1909== ===January 1909=== ====1 January 1909==== Rev. [[Social Victorians/People/Ayton|W. A. Ayton]] died (Howe 85 10-11). ===June 1909=== Summer 1909: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?). == Bibliography == #"Calendar for the Year 1900." Jumk.de Webprojects. https://kalender-365.de/public-holidays.php?yy=1900. Accessed November 2023. #Howe == Footnotes == <references /> 99478hcbs50qgt1zmsz5fr979ad21bd 0 274143 2691775 2675320 2024-12-13T10:22:40Z Elominius 2911372 Trash bin (recycle bin) introduction 2691775 wikitext text/x-wiki This lesson teaches the management of digital files and directories. While there exists no ideal method of file management, this resource documents possibly helpful practices and inspiring ideas. This lesson was created under the impression that the average computer and mobile phone user still struggles to keep track of files in the long term despite of all the tools at their disposal. == Descriptive file names == Every file manager<!-- One can safely assume that just about every file manager has this feature. If a file manager without a renaming feature happens to exist, feel free to change it, but it is very unlikely. --> has a "rename" feature, letting you change the names of files and folders. Don't just let this essential feature "rust in the junk yard". Utilize it to make files easily searchable. For example, add a short description after a picture or video file name. If you recorded using a [[smartphone]], you can rename the file on the device itself right afterwards. For example, * <code>VID_{{#time:Ymd_His}} trampoline jumping.mp4</code> It is recommended to add the short description after the file name rather than before, to prevent interference with alphanumerical sorting. Even though file managers can still sort items by date and time, files' time stamp attribute might not always be retained throughout transfers, for example when transferring files through [[:w:Media Transfer Protocol|MTP]], between a smartphone's internal memory and memory card, or uploading files to a cloud storage service. A time stamp at the beginning of the file name allows alphanumerical sorting to act as chronological sorting even if the date and time attribute is lost. Not every single file needs to be given a name manually, since that would take much effort. If your file manager lacks range selection and bulk renaming, meaning the selection and renaming of many items at once, adding a descriptive file name to the first file of similar files will still facilitate finding those files, since the title of the first file in the group covers the remaining files. For example, all four of these imaginary files depict strawberries, but only the first was manually named: * <code>IMG_20230914_090112 strawberries.jpg</code> * <code>IMG_20230914_090116.jpg</code> * <code>IMG_20230914_090119.jpg</code> * <code>IMG_20230914_090124.jpg</code> == Directories == Make use of directories. Categorize files into logically structured directories to facilitate finding them later. Some inexperienced computer users might have a habit of just throwing files into the root directory of their device such as a flash drive. With an increasing count of accumulating files, they would become increasingly difficult to find, especially multimedia content which can not simply be searched for text strings using tools like `grep` but only through metadata. Create a mind map by asking yourself where you would look for these files whenever necessary. Below are two example directory structures for video projects, where the project files of the first project are directly in the project folder, while those of the second are in a subfolder; a matter of preference. <blockquote> * Video projects ** Example project 1 *** Assets ** Example project 2 *** Project files *** Assets </blockquote> Structure files in the order your mind recalls them. For example, the type and source of files is more memorable than the date. For more examples of directory structures, see the [[/directory structures]] sub page. Feel free to add examples to it. For directories where a sufficiently descriptive directory name would be considered too long, consider creating a descriptive text file inside named <code>description.txt</code>, <code>info.txt</code>, or similar. Comments about specific files may be noted in a text file with <code>.meta</code>, <code>.meta.txt</code>, or a similar suffix appended to its name. If files within the same category have wildly varying sizes, such as command line outputs, you might want to move files beyond a size threshold such as 1 MB or 5 MB into a separate folder whose name can be the same with an added "-large" suffix. This allows minimizing the size of compressed snapshots of the directory. Avoid putting sub-directories in directories populated with many files, since they can become a nuisance when wanting to view the lastest file whereas the file manager lists sub-directories at the top, or otherwise be difficult to find between the files when needed. Due to the slow loading times of the [[:w:Media Transfer Protocol|Media Transfer Protocol (MTP)]] which is commonly used to connect mobile devices to desktop/laptop computers, and due to the possibility of file managers trying to determine file types by reading internal file information,<ref>''[https://github.com/linuxmint/nemo/issues/1907 nemo significantly slower opening folder content #1907 - GitHub]'' ([[:w:Nemo (file manager)|Nemo file manager]] reads the [[:w:Header_(computing)|header]] of each file with unknown [[:w:Filename extension|extension]]</ref> it could make navigating more difficult. Therefore, it should be considered to create a separate folder that only contains subfolders, such as <code>Downloads_2</code>. You don't need to spend much effort coming up with good names for files and folders, as you can change them later at any time if you happen to come up with a better idea. == Revision history == When spending much time on a project, write-up, email draft, etc., consider saving a revision history, by selecting "Save as" (also accessible through <kbd>Ctrl</kbd>+<kbd>↑Shift</kbd>+<kbd>S</kbd> keyboard shortcut on some software) and changing the file name through numbering or timestamping, or alternatively creating renamed copies externally using a file manager or command terminal. For possible file naming schemes and variations of numbering and time-stamping, see [[File naming]]. This enables you to revert to an earlier revsion in case of error, prevents total loss in case of failed writes caused by a power outage or software crashing, and later facilitates comprehending the work progression, while only consuming marginial space compared to common data storage, and being efficiently [[Backup#Compressed_archives|compressible]] due to redundancy. A new revision does not necessarily have to be created upon every saving, but whenever the changes since the last revision are major enough at your discretion. Optionally, a short comment summarizing the changes can be added to the file name. For programming code, suffixes like <code>-stable</code> and <code>-unstable</code> may be added after the time stamp, for example: <code>UserScriptName-revision-{{#time:YmdHis}}-stable.js</code>. Separating the number or time stamp allows convenient double-click selection in the file saving dialogue. Revisions can optionally routinely be moved into a separate subfolder. == Dumping ground == For files and folders you are unsure where to put, consider creating a directory on your device named <code>dump</code>, <code>sandbox</code>, or similar. You may wish to categorize those into text, compressed archives, drawings, or by whichever task, or dated [[file naming|folder names]] such as <code>{{#time:Y-m-d}}</code>. When managing files and directories in a command line terminal, an item can temporarily be given a simple and short name such as <code>1</code> to facilitate typing in the commands, that will be changed shortly after. Examples are: ; Output from multiple commands into one log file: <syntaxhighlight lang=sh> $ ls [path] -alR >>1 # Write names and attributes of that folder's content into the file "1". $ find [path] >>1 # Write a list of bare file paths into the file "1". $ mv -n 1 [desired file name] # Rename "1" to desired file name. "-n" prevents accidental overwriting. </syntaxhighlight> The above command could also be done in a single line with <code>(ls ''[path]'' -alR; find ''[path]'') >>''[desired file name]''</code>, but the former might be preferred to save time if the target directory contains a high number of files, as the former command starts immediately after pressing <kbd>⏎ Return</kbd> (also known as "Enter"), or when one wishes to enter commands before thinking of an output file name, while the latter requires typing in the whole command first. ; Moving files from multiple sources into a directory: <syntaxhighlight lang=sh> $ mkdir 1 # Create a folder named "1". $ mv -n *.mp4 *.mkv 1 # Move mp4 and mkv files into it. $ mv -n 1 Videos-$(date +%Y-%m-%d) # Renaming folder to intended name. </syntaxhighlight> Should it be necessary to print out the absolute (full) file path of a file for, for example, quick copying to the clipboard, use the command <code>readlink -f ''[target file]''</code>, and for information about the current mount point, use the <code>findmnt -T .</code> command. == Finished tasks == Files whose task has finished, such as a posted message that was initially drafted offline into a text file, can be moved into a subfolder of the current directory named <code>done</code>, or alternatively in one big shared folder for this purpose. == Storage types == Auxiliary data which is frequently accessed can reside on the operating system drive, on its own partition or a separate one. This especially applies to portable computers and operating systems installed on external media such as a USB flash drive. === Secondary and external storage === On desktop computers, large files can reside on a secondary large hard disk drive or solid state drive, the former of which costs less per space. For laptops, stationary hard drives located at home or portable external hard drives or solid state drives can be used. While working with a laptop as passenger in any moving vehicle such as a bus or tram, solid state memory is preferrable due to sturdiness, as hard disk drives do not like constant physical movement due to mechanical wear. If portable storage necessary, files can be stored on a constantly inserted memory card, which does not compromise ergonomy as they don't physically protrude, or at most a little. In addition, it can be occasionally removed when the data is needed on a different device. SD cards with 1 TB exist since at least 2017, though are expensive.<ref>[https://www.engadget.com/2016-09-20-sandisk-1tb-sd-card.html ''SanDisk outs the 'world's first' 1TB SD card''] (2016-09-20)</ref> A home server adds the convenient benefit of access from all devices at home, and even through the internet if set up by the user ("private cloud"), but typically has longer latency times (access delays) than physically attached storage, and lacks mass storage access that may be necessary for some programs to work properly. In such a case, the files would have to be downloaded first and worked with locally, and uploaded after finishing. === Incubate work on flash storage === If, for example, your computer has a setup with solid state drive for operating system and hard disk as expansion storage, a project may be worked on on the flash storage, and changes can be applied to the hard drive at the end of a day. If the hard drive is set to spin with no or a long timeout such as an hour or more, this may not be as necessary, but for short timeouts, frequent spin-ups would cause mechanical wear and tear in addition to annoying delays. Using flash memory can be particularly useful on battery-powered laptops for power efficiency, though mechanical hard drives are increasingly being usurped by solid state memory in laptops, mainly due to physical robustness, as hard drives on laptops have long served for cost saving, whereas solid state memory is becoming increasingly affordable since the mid-2010s, though external USB hard drives may be used on the go. === Hard drives' purposes === After purchasing a hard drive, choose whether to dedicate it to either auxiliary storage or archival ([[wikipedia: cold data|cold storage]]). Use as auxiliary storage such as for a workstation or server demands the drive be in constant operation, which wears it down over time, making it unsuitable for long-term archival. Use as archival storage demands only sporadic (rare) operation to add or retrieve data, which induces far less mechanical wear. Data retrieved from an archive drive should be copied over to auxiliary storage to avoid needless wear on the former whenever that data is recalled again in near future, as can be expected, and in order to further duplicate files that have proven to be useful, as archival is commonly done with an uncertainty of which files will be useful in future. A functional archive drive may later be repurposed for auxiliary storage after moving the archived data over to a new device. Make a list of data you wish to retrieve the next time you access your archive drive. If you can afford it, mirror the archive to a readily available hard disk or network-attached storage. If your live hard disk or network attached storage runs out of disk space, delete the files that were not accessed for the longest time if they are already backed up somewhere else. To find them, file managers are equipped with the ability to sort files and directories by "last accessed" (or similar) in either direction. === Labels === Consider physically labeling your storage media to facilitate finding data. For example, a date or a short summary of the contents can be written on a label sticker that is applied onto the casing of an external data storage device. For optical discs, a disc marker (also interchangeably known as "CD marker", "DVD marker", etc.) can be used to write notes directly on the disc. If more space is necessary, label stickers on its containing [[:w:Optical_disc_packaging#Jewel_case|jewel case]] or spindle can be used. Many jewel cases include a paper sheet for notes. Do not attach stickers directly onto the disc, as they could disintegrate during high rotation speeds, risking damage to the optical drive's internal components. It is also recommended to change the device's file system label to a year or a short summary of the contents so it can quickly be seen in the file manager's device list (usually a side bar on the left) when plugged in. On data storage devices whose life expectancy appears to be nearing its end, as indicated by [[:w:Self-Monitoring,_Analysis_and_Reporting_Technology|S.M.A.R.T. data]] on hard drives and performance loss on solid state memory. an "expired" or "EOL" (end of life) label can be added. Such devices should at most be used for temporary purposes such as testing. == Data storage size == It is recommended to get a larger storage capacity than one intends to use. This leaves some spare room in the case that more data than expected is created, such as vacation video recordings. On portable devices such as mobile phones and digital cameras, a larger storage capacity lowers the required frequency of file transfers into ones archive. Another benefit of larger storage media is that more data can be readily accessible at a given time, and a needed file can be found faster since fewer devices have to be searched. However, a central file list and [[#Labels|labeling]], as described in other sections, can facilitate finding files as well. If your storage device has, for example, 128GB, it realistically means 100 GB. Not only because of reserved space from file system overhead and possibly operating system files, but because of those 128 GB, the end is less useful, since one might want to put files there that are slightly larger than the remaining space, and one would like to have all those files in one location to avoid confusion and the annoyance of changing media. In other words, if one would like to put 11 GB where only 10 GB are available, those 10 GB are as little useful as 0 GB. == Partitioning == User data may be stored on the same or a separate partition as the operating system. The benefit of a separate partition for user data is that possible file system corruption on the operating system partition would not spill over to user data, though modern file systems such as ''NTFS'' and ''ext4'' protect themselves from damage by [[:w:Journaling file system|journaling]], which allows the file system driver to recover quickly after an unexpected termination of write access caused by an operating system crash or unexpected removal. Infrequently accessed files that may be necessary in near future can be stored there as well, or moved to a secondary drive if their size is significant. Another benefit of a separate user data partition is the smaller backup size of the operating system partition and facilitated recovery in case of a malfunctioning operating system where other means of repair have failed or would be too difficult. Because operating systems are subject to corruption and can at worst become unbootable, it is good practice to [[backup|back up]] their partition regularly into a disk image. A smaller operating system partition can be imaged more quickly and the routine induces less wear on the backup media. Should the operating system malfunction, it can be imaged and then restored more quickly from the functional previous disk image, with less work to merge desirable changes since the last backup. == Packing and archival == Perpetual streams of new files such as web downloads, photos and videos from digital cameras and cell phones, and screen captures can be packed by renaming their parent folder into a uniquely identifying name, such as with date stamp: <code>Camera-{{#time:Y-m-d}}</code>, after which they can be moved to an archive drive at the next backup appointment. The folder's name may also contain a location, device type, and/or short description. If packed more than once on the same day, time stamps or part numbers can be added to the names, or the files can be merged into the same directory. If you no longer intend to change the contents of an archived folder, a file count and byte size can optionally be added to the name to provide a quick overview in file lists and to facilitate verifying whether all data has been transferred without having to navigate back to the source device and wait for loading to view the folder's byte size. An example name is <code>Camera-{{#time:Y_m_d}}-12147367878b-2093items</code>. It is recommended to use the exact byte number to prevent confusion between file size units that are powers of two (KiB, MiB, GiB) and powers of ten (KB, MB, GB). Alternatively to renaming on the source device, the directory with uniquely identifying name can be created on the archive drive first, and files can be moved there out of the source directory. When to pack files is end users' decision, though it is recommended to do so before exhausting free space on the source device. Renaming is not as necessary if new folders are created automatically, like [[:w:Digital_camera#Directory_and_file_structure|some digital camera firmwares do per 999 or 1000 pictures]]. All filled folders can be considered eligible for archival. Larger storage space provides the benefit of more buffer until the next file transfer becomes necessary to clear space, thus it needs to be done less frequently. Individual files needed for a specific purpose such as an impending project can be copied or moved into a separate directory. == Write protection == Write protection may be desirable to defeat the fear of accidental modification of data when not desired by the user. A simple way of achieving write protection in Linux-based operating systems is to mount or re-mount a device or partition as read-only with this command which requires superuser privileges: <code>mount -o remount,ro ''[device or mountpoint]''</code>. If write-protection is not supported by the operating system, an SD card with write protection switch feature can be used. The switch relies on the SD card reader to obey it and deny writing access to the operating system. Some memory card readers, both built-in ones and USB adapters, might not obey the write protection switch. Another way to achieve write protection is finalized write-once optical media or a read-only optical drive with insufficient laser beam power to write data, as described in {{section link||Sensitive environment}}. == File listing == Searches within file lists inside a text file are significantly faster than searches through a file system. See [[Data_recovery#Create_a_file_list|this guide]] on how to create file lists. === File index === As explained in {{section link|File_puzzling#Orphaned_directories}}, some file systems store directories that comprise file paths as "linked lists", meaning distributed over the entire space rather than one index of "nodes" at the beginning, which has both benefits and disadvantages, the latter of which is slower file searching. A searchable file index stored in a text file named "index" created using the <code>find >index</code> command can facilitate finding files, as it contains a list of paths to all files at one place. The index can be updated by running the command again to overwrite the existing one. If the working directory is not the root of the file system, it should be changed to it or the paths need to be specified. The "index" file can be searched with ease using <code>grep -i "searched file name" index</code>, which is typically much faster than directly searching the file system. <code>-i</code> may be left out for case-sensitive search. These commands have additional options, but these are outside the scope of this section. == Time stamp preservation == Some methods of file transfer, such as copying within/onto mobile phone storage, the <code>cp</code> command without activated <code>-p</code> ("preserve") option, and a directory on Unix/Linux not owned by the current user, might discard date and time stamp file attribute(s), resetting it to the current time. To preserve last-modified time stamps over FTP, downloading is preferred, as uploading while preserving it requires both client and server to support the [[:w:FTP#Additional_commands|MDTM (''Modify Fact: Modification Time'') command]], which it is not widely. == High numbers of small files == With an increasing number of files, file searches slow down. High numbers of small files also restrict portability, as they demand more file operations for file transfers, slowing the process down.<ref>Related: [https://www.quora.com/Why-is-copying-1-000-1MB-files-so-much-slower-than-copying-1-1GB-file-given-that-the-same-amount-of-data-is-being-copied/answer/Franklin-Veaux Answer on Quora to "''Why is copying 1,000 1MB files so much slower than copying 1 1GB file, given that the same amount of data is being copied?''" by Franklin Veaux, on January 14th, 2020]</ref> Additionally, higher cluster sizes in combination waste more space to [[File Systems#Cluster_size|cluster overhead]] (unused reserved space). If you happen to have a high number of currently unneeded small files, such as tens of thousands, consider packing them into one big archive file for improved portability. Compression may be considered where efficient, such as in human-readable text files and code, and/or where more necessary, such as online file sharing. Compression ratios of 100 may be achievable by strong compression algorithms on text documents and code. However, it should be taken into consideration that damage magnifies enormously over compressed archives, as demonstrated in {{section link|Backup#Compressed archives}}. Therefore, it is recommended to store compressed archives on at least two devices. Text inside compressed archives can be searched through directly without extracting using tools such as <code>zgrep</code>, <code>xzgrep</code>, <code>bzgrep</code>, and for 7-Zip, <code>7z e -so -bd ''[path]'' |grep ''[query]''</code>. If frequent modification of small files is necessary, an alternative to packed archive files (such as <code>.tar</code>, <code>.zip</code>) are file system containers (''virtual disks'') with a small cluster size. File system containers are manually generated disk image files that can be mounted like usual drives. == Avoid exhausting the operating system's partition == Exhausted space storage should be avoided, especially on an operating system partition, as it could lead to bogus behaviour by software not designed to handle such condition, or other unwanted behaviour. For example, a failed write while saving could blank the target file, causing the loss of work and reset of configuration. A web browser might [[Backup#Browsing_history|automatically delete early browsing history entries]] to make space for new. Even seemingly basic features such as command line parameter completion could malfunction.<ref>{{cite web |url=https://unix.stackexchange.com/questions/277387/tab-completion-errors-bash-cannot-create-temp-file-for-here-document-no-space |date=2016-04-18 |access-date=2022-02-10 |title=Tab completion errors: bash: cannot create temp file for here-document: No space left on device |website=Unix & Linux Stack Exchange }}</ref> On an operating system partition, keeping a safety margin of free storage such as 5% at any time is recommended, and at least 1% on secondary expansion storage. On archival media, a controlled exhaustion of space is less critical, though the readability of the final written files should be verified. Should you still find yourself with 100% exhausted space, first seek few megabytes of files to move out, perhaps temporarily, to improve system stability. This allows you to take time to calmly search for more and/or larger files to move out. == File system repairs == Tools such as <code>CHKDSK</code> on Microsoft Windows and <code>fsck</code> on Linux promise repairing damaged file systems. Logical file system errors may be caused by unexpected power outages or unpluggings. Be careful with file system repairs. It is recommended to back up any device (either to a full-disk image or by copying all files) prior to running one, in order to be able to revert with ease in case of unwanted collateral damage. Detected file system errors may be caused by incompatibile file names across operating systems. For example, Linux allows characters in file and folder names in NTFS that Windows considers invalid, such as a colon (<code>:</code>) and a pipe character (<code>|</code>), as well as case-insensitive file names. Upon detection of invalid characters, the <code>CHKDSK</code> tool moves and renames such items, which leads to the loss of file names and paths. In particular, <code>ChkDsk</code> moves files and folders with invalid names into a directory located at the file system's root named <code>found.000</code>, and renames them to generic names like <code>file00000000.chk</code> and <code>dir_00000000.chk</code>, where the number is hexadecimal and incremented. == Disk usage analysis == [[File:GNOME_Disk_Usage_Analyzer_3.32_screenshot.png|thumb|Disk usage analyzers facilitate finding directories with the largest content size. Some illustrate the directory structure graphically.]] Disk usage analyzers calculate the size of directories on any selected path, allowing the user to easily discover directories which occupy the most space. Large folders not currently needed can be moved over to an archive drive, which clears the most space on the source device. Popular tools for desktop operating systems include ''[[:w:Disk_Usage_Analyzer|Baobab]]'' for Linux (pre-installed on some popular distributions) and ''Xinorbis'' for Windows, both with sophisticated graphical user interfaces. Linux is also equipped with the command-line tool <code>du</code>, which allows outputting results directly into a text file. For mobile (Android OS), [[:w:ES_File_Explorer|ES File Explorer]] is equipped with such functionality, though that application has been subject to controversy and has developed into adware. <div style="float:none; clear:both;"></div> == Deduplication == Several tools to automatically deduplicate files exist: rdfind, fdupes, jdupes, rmlint, dupeGuru, FSlint. Storing one duplicate of files may be desirable in certain situations, such as [[Backup#Compressed_archives|compressed archives]] intended for long-term preservation, where even the slightest damage can render any data after the point of damage unreadable. For accessing the same file from different locations, [[:w:Hard_link|hard link]]s or [[:w:symbolic link|symbolic link]]s can be used. You may want to deduplicate files across two different storage devices without accidentally deleting any files that do not exist on the storage device you want to keep the files on. A manual deduplication may be necessary if you wish to free up space from a device after copying files to a different device or after creating an archive file from those files without accidentally deleting any files that were not copied or archived from the source device, or after a file copying operation was interrupted and you need to clean up without deleting any files that were not copied. The quickest way to accomplish this would be to compare the total count and size of those files in both the source and the target device by selecting the files and opening the "Properties" window. If the byte counts match, the source folder can be safely deleted. However, if the selection contains subdirectories too, differences in file systems mean that the total size might not match due to differences in how directories are stored. Some file managers also don't show the exact byte count. If you want to deduplicate files between two different computers or between a computer and a smartphone, an additional problem is that file managers count files and folders differently. Some file managers only show the number of "items", meaning both files and folders, where as others may only show the number of "files" without counting the folders. If possible, use the same file manager on both devices, and one that shows the exact byte count. A more sophisticated way to manually deduplicate files is creating a temporary script that contains a list of files to be deleted. In Linux, this can be accomplished first changing the working directory to the target folder (where the files are supposed to stay) using <code>cd</code> and then creating a temporary script by running: <syntaxhighlight lang=sh> # list files to be deduplicated find -type f |sed -r "s/(.*)/rm -v '\1'/g" >>temporary.sh # list empty directories to be removed find -type d |sed -r "s/(.*)/rmdir -v '\1'/g" >>temporary.sh # script deletes itself after work done echo "rm temporary.sh" >>temporary.sh </syntaxhighlight> No files have been deleted up to this point. Only the disposable script to do the work has been created. This script contains a list of files to be deleted from the source device, but the list is generated from the target device to avoid containing the name of any files that have not been copied. It is recommended to use a temporary script rather than throwing around commands involving an <code>|xargs</code> pipe, since running <code>|xargs</code> commands in combination with deletion is dangerous since you don't see a list of files before starting the deduplication process. The <code>-v</code> flags are optional and serve to later show the names of the files deleted in the deduplication process in the terminal. Now move the <code>temporary.sh</code> file to the original directory on the source device from which you want to delete the duplicate files. Then change your terminal's working directory to that directory using <code>cd ''[path]''</code>. Before running the script, open it in a text editor and glance over it to verify that it doesn't contain any files you don't intend to delete. Then run it using <code>sh temporary.sh</code>. Do not run the script before changing your working directory to the same directory the script resides in, because the script contains relative paths rather than full (absolute) paths, so the path of your terminal's working directory will be presumed to be infront of the relative paths in the script. == Temporary folder == Programs may use temporary folders such as <code>/tmp/</code> and <code>~/.cache/</code> on Linux and <code>%temp%</code> on Windows to store data such as preview thumbnails and data from the web to reduce loading times. Should the way your operating system or file manager handles its [[:w:Temporary folder|temporary]] or [[:w:Trash (computing)|trash folder]] not suit your needs (e.g. if retention span is unchangeable or limited), you may wish to manually operate such in your user home folder (e.g. <code>~/tmp/</code>, <code>~/trash/</code>). Some software may allow changing the path for the temporary folder. Furthermore, such folders can be used for frequent short-term backups of projects, which can be deleted when free space becomes necessary. In comparison to a traditional "Trash" folder implemented by the operating system or file manager, files can be opened directly as usual, whereas a file manager may disallow directly opening files sitting in a "Trash" folder and show file properties instead, as Windows Explorer does, whereas, for example, the ''Nemo'' file manager for Linux allows directly opening files located in the trash directory. == Trash bin (recycle bin) == When deleting files, you might want to consider using the [[:w:Trash (computing)|trash bin]] (or "recycle bin" and other names) feature, which stores files in a temporary location until they are automatically deleted. This allows reversing unwanted deletions, for example if you selected a file or folder you didn't mean to delete. This lifts the emotional burden and carefulness during deletion, since you know you can undo the deletion for the time being. Without it, deletions are "walks on eggshells", metaphorically speaking. A recycle bin also makes file moving (with copying and deletion from source) safer, because during the deletion step, you have to be less careful to select the exact same files and folders that you moved. If your trash bin does automatic deletion after a certain time, you don't have to bother with cleaning it up later anymore, while you can still bring back the file if necessary in the near future. The files still take up space on your device while residing in the trash bin, but it doesn't matter if you have enough space free. == Mobile == === Memory cards === [[File:Huawei_U8950D_no_cover.JPG|thumb|right|Smartphone with inserted memory card (located below camera lens)]] Some smartphones and tablet computers allow the expansion of storage capacity using memory cards, typically MicroSD, which significantly facilitates file management and is user-friendly. Memory cards can be re-used immediately between devices without need for file transferring, and data stored on the memory card is not at the risk of mobile devices' technical defect, as it can be ejected, after which data can be retrieved externally. Mass storage access from an external computer also may allow [[data recovery|recovering some files]] imminently after a deletion accident caused by bogus software<ref>[https://www.reddit.com/r/Android/comments/16e6kc/i_just_deleted_a_random_folder_in_my_internal/ Users report Android devices' entire internal user storage being deleted instantly, caused by poor software design – "''I just deleted a random folder in my internal storage and it wiped my internal storage. What the heck just happened?''" – Reddit.com/r/Android (2013-01-11)]</ref> and/or human error. For huge file transfers, ejecting the memory card and directly transferring to the PC through mass storage may save time compared to MTP (media transfer protocol) through the phone or tablet, as the latter does not handle high counts of files within a directory well. Additionally, memory cards can immediately be reused in a different device without lengthy file transfers. USB-OTG (''On The Go'') may be used as well, connected through an adapter directly to the mobile device, though it might not [[#Time_stamp_preservation|preserve a date and time attribute]]. Tablet computers with desktop operating system are widely equipped with at least one default-sized USB-A port. Additionally, using a memory card takes stress off the device's non-replaceable internal memory, preserving its limited rewrite cycles, which is especially beneficial for repeated heavy tasks such as high-resolution filming and mobile FTP server hosting. <div style="float:none; clear:both;"></div> === Between computer and mobile === ==== Media Transfer Protocol ==== File management on mobile phones and tablet computers with mobile operating systems is more restricted than on desktop/laptop computers and mass storage devices such as USB sticks and memory cards, as the media transfer protocol (MTP), which is used to access files on a mobile device from a computer, lists files slowly, which is problematic for loading directories with high counts of files. As such, it is recommended to manage such directories on the device itself. If transfer between a desktop/laptop computer is desired, handle those files in a little to unpopulated directory. MTP file listing can be sped up by not loading preview thumbnails. Depending on file manager used, this may be done by deactivating preview thumbnails in settings or choosing the "detail" view mode, where files appear in a list instead of a grid which foregoes the loading of preview thumbnails. However, files can be dropped into a directory without opening the target directory, by pasting them through the right-click context menu. A benefit of MTP is it not being prone to file system corruption as a result of unexpected removal, meaning without being "safely unmounted" through the client operating system (i.e. desktop / laptop computer), as it operates through an abstraction layer and the file system is controlled by a driver on a battery-powered host device (i.e. smartphone / tablet). Only a selection of files, no directories, should be moved away from the device, because users have reported files on MTP not being listed properly.<ref>[https://android.stackexchange.com/questions/136839/files-show-up-on-nexus-5-but-not-in-windows-7 ''Files show up on Nexus 5 but not in Windows 7'' – Android Stack Exchange – February 11th, 2016]</ref><ref>[https://android.stackexchange.com/questions/46315/not-all-files-are-visible-over-mtp ''Not all files are visible over MTP'' – Android Stack Exchange – May 29th, 2013] (209,502 views as of September 21st, 2022)</ref><ref>[https://stackoverflow.com/questions/13737261/nexus-4-not-showing-files-via-mtp ''Nexus 4 not showing files via MTP'' – StackOverflow – December 6th, 2012] (67,504 views as of September 21st, 2022)</ref> If a directory is moved away from the device, the computer might delete it from the mobile device without all content having been transferred away. Instead, it should be copied, and the byte size be compared on both the computer and the smartphone itself, where a match indicates a successful transfer, meaning the directory can now be deleted from the mobile device. The only exception where moving folders out is safe is when the number of files within is overseeable, i.e. less than ten, where all files are clearly listed in the computer's file manager. Windows Explorer additionally displays files while listing is in progress, which can be of use when moving files out, since the loading of the file list can be interrupted to allow moving out the displayed files chunk-wise, reducing the number of remaining ones each time. Transferring files onto the device through MTP [[#Time_stamp_preservation|may dismiss their date and time attribute]]. If a file has newly been created on the smartphone while it was connected to the computer through MTP, the computer's file manager could potentially misreport the file size as too small due to having loaded the directory listing in a moment where the file was not complete. Moving the file away from the phone could cause it to be truncated (incomplete) on the target path while being deleted from the source, since the file manager might wrongfully assume that the file has been fully transferred while it hasn't. ==== File Transfer Protocol ==== An alternative to MTP is FTP (file transfer protocol) through ethernet. On the desktop computer, a dedicated and sophisticated FTP client such as FileZilla (open-source) may be used to handle high numbers of files, though FTP is widely supported by file managers and web browsers. FileZilla does not support moving files out of an FTP server, meaning downloading and deleting automatically, whereas moving within a server is supported through the standard rename command. If the intention is moving files out of an FTP server, the highlighted selection of files on the server needs to be deleted after the transfer after verifying that all files have been transferred successfully, meaning no new entries in the "Failed transfers" list. To get a peace of mind that the selection was transferred successfully, try downloading it again while skipping existing local files. If no new files are downloaded, this ensures all files have already been transferred. This might apply to other software as well. FTP server applications for mobile devices may handle file listing differently. Some do not report the year of the file, only day and month, causing the FTP client to insert the current year for files except those last modified at a later time of the year than currently, for which the previous year is inserted instead. Another distinction between FTP server apps is whether they list file and directory names starting with a dot, which is considered ''hidden'' in the Unixverse (i.e. on Unix and Linux-based operating systems, which includes Android OS, the most popular mobile operating system). FTP server applications typically allow the user to select a specific directory to share, rather than the entire storage. This feature has been [[:File:Android extended MTP options concept.png|proposed for MTP]], but never implemented there so far. Two open-source FTP server apps for Android OS are the integrated FTP server of "Amaze File Manager", and the more sophisticated "primitive FTPd", only the latter of which reports files and folders with names starting with a dot. Alternatively, files may be uploaded vice versa from the mobile device to an ethernet FTP site served by a home computer, though as of 2021, no mobile file manager's FTP client supports [[File management#Time stamp preservation|preserving files' date and time stamps]] upon uploading. ==== Handling invalid file names ==== Since the most popular mobile operating system, Android OS, is Linux-based, it supports characters in file names that are unsupported by the most popular desktop operating system, Windows, and by some file systems. These characters include a colon (<code>:</code>), a back slash (<code>\</code>), a vertical pipe (<code>|</code>), a question mark (<code>?</code>), and an asterisk (<code>*</code>). Additionally, file names are case-sensitive, meaning files named "file" and "File" and "FILE" can co-exist within a directory on Linux and Android OS, but not on Windows. While some mobile phone apps disallow the creation of such files and automatically replace characters Windows considers invalid with a substitute character such as an underscore (<code>_</code>), other apps might have created files with names containing aforementioned characters. When copying or moving such files from a mobile device through MTP (Media Transfer Protocol) using Windows, all characters before and including the last invalid character are discarded from the file name on the target location to make the file openable. Therefore directories on a mobile device which contain such files will need to be moved out in two passes to retain the file names. First, an [[:w:archive file|archive file]] such as a ZIP file should be created on the mobile device which can then be moved out before the files themselves can be moved out. The archive file can optionally be created after isolating files with invalid characters in a separate directory to consume less space, though that can be a difficult task on a mobile device due to limited file management software and users' infamiliarity with the Linux terminal which can be accessed on Android OS through third-party applications such as Termux or Jack Palevich's "Android Terminal" app. Another option would be to only create the archive file and not transfer the bare files to the archival media. However, this requires files to be extracted before being opened, which adds a delay for larger files and does not allow for preview thumbails without extracting the entire archive file. === On-device management === Additionally, file access on the most popular mobile operating system has been restricted significantly over time, and to varying degrees per storage type (internal, memory card, and USB-OTG). Such restrictions affect third-party applications installed by the user, including file managers. Pre-installed file managers are usually unaffected, though these tend to be functionally restricted, such as lacking range selection, where only two entries need to be tapped for all inbetween to be marked. [[:File:Android extended storage options concept.png|Options to deactivate these restrictions at user discretion]] were not officially provided, leaving so-called ''rooting'' as the only possibility of bypassing them. This is a process in which the user unlocks administrative access over the operating system. The operating system vendor claims aforementioned file access restrictions to serve user security, though them being a cloud storage vendor as well suggests a commercial interest that conflicts with end users' desire of freedom, and simultaneously may encourage users to unlock ''root'' access, which is against vendors' recommendation, and where inexperienced tampering can lead to malfunction. == Other ideas == === Archival queue === New files from portable devices which are currently unneeded can be moved into a buffer directory of files ready for archival, which means moving them to a large and stationary hard drive at the next connection to the computer. External flash storage such as USB sticks and solid state drives can also be used to store data for, for example, the duration of a trip or vacation, where they can be moved to an archive hard drive when arriving at home. === Temporary redundant retention after archival === Files that have already been moved to a larger stationary archive drive may be redundantly kept on the smaller portable data storage such as a mobile phone or USB stick, but in a directory in which any file is eligible for deletion, such as at space exhaustion. This would serve as a short-term backup, which could be retrieved from in case anything goes wrong with the archive drive prior to it getting backed up itself. This increases file fragmentation on the portable device, though that does not noticably affect performance on flash storage. === Partition for small files === If your computer setup has no secondary drive and/or partition, you may create a small partition (e.g. 4 GB) with a low cluster size for more efficient storage of small files. Additionally, if space storage happens to be exhausted on the main partition, with software arbitrarily attempting to write to it, files can still be added on the secondary partition without interference. === Sensitive environment === Inside sensitive environment, data may be exchanged through rewritable optical media such as DVD±RW and BD-RE, as these use external storage controllers, making the media itself unable to contain malicious hardware such as so-called ''rubber duckies'' used to simulate keystrokes from a USB keyboard. Additionally, finalized write-once media and/or read-only (''ROM'') optical drives can ensure write protection where necessary, for example in a malware-infested environment. === Copies of description files === Description files such as text files describing the contents of a folder may be stored both inside a folder and a copy in a central location with other description files for easier discovery. The name of that file should contain the name of the folder it is describing. It is recommended to store such a description file inside the folder it is describing rather than along with it in the parent directory, to make it easier to find if the items in the parent directory are not sorted alphabetically, since a file managers might separate files and folders regardless of which sorting method (alphabetical, by size, by name, by last modified) is chosen, which would make the description file not appear next to the folder it is describing. == Observations and tips == === Spare directories === When space on a device or partition is exhausted, no new directories that could be helpful for organizing files can be created, such as for moving files from a highly ''populated'' directory (i.e. with many files, such as a download folder) on a mobile phone in order to skip having to open the populated folder directly through [[#Media Transfer Protocol|MTP (Media Transfer Protocol)]], which notoriously handles long file listings poorly. Prepare for such a situation by creating a reserve of spare empty directories inside one dedicated directory. The spare directories can be moved out of the reserve, and be renamed as necessary, even without space left, which allows organizing files on the go to be able to move them elsewhere immediately when arriving home. === File move behaviour === When the aim is to bring files to an archive, moving files rather than copying and deleting afterwards has the convenience benefit of acting like a check list of files, instead of creating duplicates that would later have to be sorted out without accidentally deleting non-copied files, as well as imminently clearing (freeing) space on the source device. Moving instead of copying and then deleting files also defeats the psychological barrier that may come from the deletion step, as it feels like a destructive action even though it leads to the same result as moving between storage devices. Another barrier is the uncertainty of having inadvertently selected any file not copied. When moving files, some file managers may delete files individually after transfer, while others only delete selected files from source only after the last file has finished transfering. Windows Explorer uses the former method for mass storage devices, but the latter with Media Transfer Protocol. The Linux file manager ''Nemo'' always uses the former method. With the latter method, any interruption would cancel the file transfer without having freed up any space on the source device. === Escape auto-closure === Some file managers such as Windows Explorer close themselves when detecting the removal (unmounting and/or physical unplugging) of a storage device, while others jump to the starting directory. Some file managers might do nothing. The first case may be perceived as annoying, as it forces users to re-open the file manager and navigate all the way back to the previous directory.<ref>[https://www.sevenforums.com/general-discussion/148620-how-stop-windows-explorer-closing-removable-disks.html ''How to stop Windows Explorer closing for removable Disks!'' &ndash; SevenForums]</ref> This can be prevented by opening a different device in the file manager before unplugging. After plugging the device back in, the previously opened directory can be navigated back to immediately through the navigation history, using the on-screen {{button|&larr;}} button or <kbd>Alt</kbd>+<kbd>&larr;</kbd> on the keyboard. === Using the command line === Command-line file operations can be logged for later reference by using the "verbose" switch <code>--verbose</code> or shorthand <code>-v</code> on the <code>cp</code> and <code>mv</code> commands in Linux and redirecting the output into a text file by appending <code>>>path/to/textfile.txt</code>. On Windows, file operations are outputted by default, meaning no "verbose" switch is necessary. On Linux, using <code>| tee -a path/to/textfile.txt</code> allows visibly printing out command line output in real-time (as usual) while logging into a text file simultaneously. Autocompletion using the <kbd>↹Tab</kbd> key is widely supported on both Microsoft Windows and Linux/Unix-based operating systems and facilitates navigation by selecting file and folder names.<ref>[https://www.computerhope.com/tips/tip176.htm Use tab to autocomplete commands in the command line (ComputerHope, 2020-12-31)]</ref> === Shortcuts === Don't hesitate to use bookmarks to frequently accessed directories in your file manager and file picker dialogue (also known as "Open" and "Save as"). If they become unnecessary, they can be removed at any time. For command-line use, familiarize yourself with environment variables, as they allow quicker navigation throughout your directories. They include <code>~</code> on Linux and <code>%user profile</code> (user home directory, equivalent to <code>C:\Users\''Username''\</code>) and <code>%temp%</code> (<code>C:\Users\''Username''\AppData\Local\temp\</code>). Environment variables are also typically recognized by file pickers. Additional shortcuts for command-line use can be specified as variables in the script that runs when starting the terminal, which in Linux is typically located at <code>~/.bashrc</code>. For example, <code>$dl</code> can be set to refer to the download folder by appending <code>dl=~/Downloads</code> to that file. === Spaces in file names === Keep in mind that space characters in file and path names can become a nuisance when entering paths, creating variables, selecting text, and navigating and auto-completing in a command prompt/terminal using the <kbd>↹Tab</kbd> key. === Cleanup === "Cleanup" may refer to moving files scattered around the storage and desktop in folders to improve overview, or deleting files like duplicates to clear free space. In the former case, refer to [[/Directory structures]] and consider moving files you are unsure where to move into a [[#Dumping_ground|dumping ground]]. In the latter case, as with [[Backup#lostBackup|lost backups]], first weigh effort against price, meaning here first consider whether the time and effort spent searching files to clean up really outweighs the storage space price. It is not worth trying to search and eliminate duplicate files for the sake of it if no significant space is cleared. The time and effort necessary to do it might outweigh any benefit from the saved space. However, duplicates may be deleted for organizational purposes, such as duplicate music tracks in the same folder. Note that cleaning does not necessarily speed up the computer noticeably, except if the partition was nearly full, which should be avoided anyway, as described in [[#Avoid_exhausting_the_operating_system's_partition|§ Avoid exhausting the operating system's partition]]. Rather, closing tasks that stress the CPU and take much RAM is effective in speeding up the system. Clearing caches is mostly unnecessary and even disadvantageous, as they provide faster recall of information. For files vulnerable to damage like PNG and TIFF images, and [[Backup#Compressed_archives|compressed archives]], duplicates are beneficial for files not backed up yet. == Experiences by participants == This section goes deeper into the idea about that "the average computer and mobile phone user still struggles to keep track of files in the long term despite of all the tools at their disposal". Here participants or visitors to this learning resource can add why they need a resource such as this to learn from and/or interact with it. :''Moved to [[Talk:{{PAGENAME}}]].'' == References == <references /> == See also == * [[File naming]] * [[File systems]] * [[Backup]] * [[Data recovery]] == External links == * Comics: ** [https://xkcd.com/949/ File Transfer – xkcd #949, 2011] ([https://www.explainxkcd.com/wiki/index.php/949:_File_Transfer explainer]) ** [https://xkcd.com/1360/ Old Files – xkcd #1360, 2014] ([https://www.explainxkcd.com/wiki/index.php/1360:_Old_Files explainer]) ** [https://xkcd.com/1817/ Incognito Mode – xkcd #1817, 2017] ([https://www.explainxkcd.com/wiki/index.php/1817:_Incognito_Mode explainer]) [[Category:Information technology]] [[Category:Computer_science]] 5hpv672j370j5rh3lf0avfsm0fu6dig 0 275305 2691718 2535087 2024-12-12T23:20:13Z Yinuo Liu 2935880 address error warnings, add banner + instrument table 2691718 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{Wikipedia2|PHQ-9}} The '''PHQ-9''' (DEP-9 in some sources<ref>{{Cite book|url=https://books.google.com/books?id=f7nA6ym7EhEC&q=%22dep-9%22+depression&pg=PA23|title=Clinical Psycho-Oncology: An International Perspective|last1=Grassi|first1=Luigi|last2=Riba|first2=Michelle|date=2012-05-18|isbn=9781119941095}}</ref>) is a 9-question instrument given to patients in a [[w:primary care | primary care]] setting to screen for the presence and severity of [[w:Depression (mood)|depression]]. It is the 9-question depression scale from the [[w:Patient Health Questionnaire|Patient Health Questionnaire (PHQ)]]. The results of the PHQ-9 may be used to make a depression diagnosis according to [[w:Diagnostic and Statistical Manual of Mental Disorders|DSM-IV]] criteria and takes less than 3 minutes to complete. The total of all 9 responses from the PHQ-9 aims to predict the presence and severity of depression. Primary care providers frequently use the PHQ-9 to screen for depression in patients. == History == The PHQ-9 is the 9-question depression scale of PHQ. The PHQ is a self-administered version of the [[w:Primary Care Evaluation of Mental Disorders|PRIME-MD]], a screening tool that assesses 12 mental and emotional health disorders. The PHQ is 59-question instrument. It has modules on [[w:Mood disorder|mood]] (PHQ-9), [[w:Anxiety disorder|anxiety]], alcohol, eating, and [[w:Somatic symptom disorder|somatoform]] disorders.<ref>{{Cite web|url=http://www.phqscreeners.com/select-screener/31|title=PHQ Screener Overview|website=Pfizer}}</ref> Dr. Robert J. Spitzer, Dr. Janet B.W. Williams, Dr. Kurt Kroenke, and colleagues from [[w:Columbia University | Columbia University]] developed the PHQ in the mid 1990s and the PHQ-9 in 1999 with a grant from [[w:Pfizer | Pfizer]].<ref name=":0">{{Cite web|url=https://phqscreeners.pfizer.edrupalgardens.com/sites/g/files/g10016261/f/201412/instructions.pdf|title=PHQ-9 Screener-Phizer}}</ref> == Survey items == A patient may take the PHQ-9 in written form or be asked the survey items by clinic staff. The PHQ-9 questions are based on diagnostic criteria of depression from DSM-IV and ask about the patient's experience in the last 2 weeks. Questions are about the level of interest in doing things, feeling down or depressed, difficulty with sleeping, energy levels, eating habits, self-perception, ability to concentrate, speed of functioning and thoughts of suicide. Responses range from “0” (Not at all) to “3” (nearly every day).<ref>{{Cite web|url=http://www.phqscreeners.com/sites/g/files/g10016261/f/201412/PHQ-9_English.pdf|title=PHQ-9}}</ref> Clinicians may ask a 10th question that asks how difficult the problems that the prior questions ask about make it to function in daily life. The 10th question is not factored into the final score and clinicians may use it to gauge the patient’s opinion of the level of impairment caused by their mental health.<ref name=":0" /> == Interpretation of results == The total sum of the responses suggests varying levels of depression. Scores range from 0 to 27. In general, a total of 10 or above is suggestive of the presence of depression. Listed below are PHQ-9 totals, the levels of depression that they relate to, and suggested treatment for each level of depression:<ref name=":1">{{Cite journal|last1=Kroenke|first1=K|last2=Spitzer|first2=R.L.|year=2002|title=The PHQ-9: A New Depression Diagnostic and Severity Measure|url=https://pdfs.semanticscholar.org/de26/1882049731e262c7ba4a2e0a710cd0cc807c.pdf|journal=Psychiatric Annals|volume=32|issue=9|pages=1–7|doi=10.3928/0048-5713-20020901-06|archive-url=https://web.archive.org/web/20170324174614/https://pdfs.semanticscholar.org/de26/1882049731e262c7ba4a2e0a710cd0cc807c.pdf|archive-date=2017-03-24|url-status=dead|s2cid=17881533}}</ref> {| class="wikitable" !PHQ-9 Score !Depression severity !Suggested Intervention |- |0-4 |None-minimal |None |- |5-9 |Mild |Repeat PHQ-9 at follow-up |- |10-14 |Moderate |Make treatment plan, consider counseling, follow-up, and/or prescription drugs |- |15-19 |Moderately Severe |Prescribe prescription drugs and counseling |- |20-27 |Severe |Prescribe prescription drugs. If there are poor responses to treatment, immediately refer the patient to a mental health specialist for counseling. |} A provisional diagnosis of Major Depressive Disorder can be made by using responses to PHQ-9 questions to fulfill the diagnostic criteria of DSM-5. According to DSM-5, Major Depressive Disorder is likely if 5 or more of the 9 symptoms are present for “most of the day, nearly every day" in the past 2 weeks and one of the symptoms is depressed mood or little interest or pleasure in doing things (questions 1 and 2 on the PHQ-9). Any degree of suicidal thoughts counts toward this criteria. The symptoms must also cause significant distress and loss of function, and the symptoms must not be better explained by substance use or another medical or psychiatric condition. “Other” depression is diagnosed if there is significant impairment and/or distress in major areas of functioning, but the full criteria for any specific depressive disorder are not met.<ref>{{Cite book|title=Desk reference to the diagnostic criteria from DSM-5|last=Association|first=American Psychiatric|date=2013|publisher=American Psychiatric Publishing|isbn=9780890425633|oclc=825047464}}</ref> The PHQ-9 can be used to diagnose Major Depressive Syndrome, but Major Depressive Disorder must be diagnosed using additional clinical information (e.g. existence of past manic/hypomanic episode, bereavement, other mental disorder, effects of a medication or illness).<ref>{{Cite web|url=https://phqscreeners.pfizer.edrupalgardens.com/sites/g/files/g10016261/f/201412/instructions.pdf|title=Instructions for Patient Health Questionnaire (PHQ) and GAD-7 Measures|author=Pfizer Inc.|date=2010|website=Pfizer}}</ref> Clinicians may also use the PHQ-9 to evaluate treatments given for depression. A change of PHQ-9 score to less than 10 is considered a “partial response” to treatment and a change of PHQ-9 score to less than 5 is considered to be “remission.” <ref name=":1" /> == Validity and reliability == Kroenke, Spitzer, and Williams conducted validity and reliability tests on the PHQ-9 in 2001. [[w:Reliability (statistics)|Reliability]] and tests found a [[Cronbach's alpha|Cronbach’s alpha]] of 0.89 among 3,000 primary care patients and 0.86 among 3,000 [[w:Obstetrics and gynaecology|OB-GYN]] patients. The test-retest reliability was assessed by the [[w:Correlation and dependence|correlation]] between PHQ-9 scores obtained from in-person and phone interviews with the same patients. The correlation value obtained was 0.84.<ref name=":2">{{Cite journal|last1=Kroenke|first1=K|last2=Spitzer|first2=R.L.|last3=Williams|first3=J.B.W.|year=2001|title=The PHQ-9: Validity of a Brief Depression Severity Measure|journal=Journal of General Internal Medicine|volume=16|issue=9|pages=606–613|doi=10.1046/j.1525-1497.2001.016009606.x|pmc=1495268|pmid=11556941}}</ref> In an assessment of [[w:construct validity | construct validity]], the correlation coefficient between the PHQ-9 and the SF-20 mental health scale was 0.73. To assess [[w:criterion validity | criterion validity]], a mental health professional validated depression diagnoses from PHQ-9 scores from 580 participants, resulting in 88% [[w:Sensitivity and specificity|sensitivity]] and 88% [[w:Sensitivity and specificity|specificity]].<ref name=":2" /> == Applications == The [[w:National Institute for Health and Care Excellence|National Institute for Health and Clinical Excellence]] endorsed the PHQ-9 for measuring depression severity and responsiveness to treatment in a primary care setting.<ref name=":3">{{Cite journal|last1=Smarr|first1=K.L.|last2=Keefer|first2=A.L.|year=2011|title=Measures of depression and depressive symptoms: Beck Depression Inventory-II (BDI-II), Center for Epidemiologic Studies Depression Scale (CES-D), Geriatric Depression Scale (GDS), Hospital Anxiety and Depression Scale (HADS), and Patient Health Questionnaire-9 (PHQ-9)|journal=Arthritis Care & Research|volume=63|pages=S454–S466|doi=10.1002/acr.20556|pmid=22588766|s2cid=9086062}}</ref> [[w:Behavioral Risk Factor Surveillance System|The Behavioral Risk Factor Surveillance Survey]] (BRFSS), the National Health and Nutrition Examination Survey, the Medical Expenditure Panel Survey, the National Epidemiologic Survey on Alcohol and Related Conditions, the Medicare Health Support program, and the Millennium Cohort Study use the full PHQ-9 or a shortened form of it. The [[w:United States Department of Veterans Affairs|Veterans Administration]], [[w:United States Department of Defense|Department of Defense]], and [[w:Kaiser Permanente | Kaiser Permanente]] adopted the PHQ-9 as a standard measure for depression screening. The PHQ-9 is also the most commonly used depression measure in the [[w:United Kingdom|United Kingdom's]] [[w:National Health Service (England)|National Health Service]], which requires providers to use a depression screening instrument when treating depression.<ref name=":4">{{Cite journal|last1=Kroenke|first1=K|last2=Spitzer|first2=R.L.|last3=Williams|first3=J.B.W.|last4=Lowe|first4=B|year=2010|title=The Patient Health Questionnaire Somatic, Anxiety, and Depressive Symptom Scales: A Systematic Review|url=http://people.oregonstate.edu/~flayb/MY%20COURSES/H676%20Meta-Analysis%20Fall2016/Examples%20of%20SRs%20&%20MAs%20of%20associations/Kroenke%20etal10%20SR%20Patient%20health%20questionnaire%20scales.pdf|journal=General Hospital Psychiatry|volume=32|issue=4|pages=345–359|doi=10.1016/j.genhosppsych.2010.03.006|pmid=20633738}}</ref> Studies found the PHQ-9 is also useful for screening for depression in [[w:Psychiatry|psychiatric]] clinics<ref>{{Cite journal|last1=Inoue|first1=T.|last2=Tanaka|first2=T.|last3=Nakagawa|first3=S|last4=Nakato|first4=Y.|last5=Kameyama|first5=R.|last6=Boku|first6=S.|last7=Toda|first7=H.|last8=Kurita|first8=T.|last9=Koyama|first9=T.|year=2012|title=Utility and Limitations of PHQ-9 in a Clinic Specializing in Psychiatric Care|journal=BMC Psychiatry|volume=12|pages=1–6|doi=10.1186/1471-244X-12-73|pmc=3416649|pmid=22759625}}</ref>. Studies have used the PHQ-9 to study patients with [[w:Diabetes mellitus|diabetes]],<ref>{{Cite journal|last1=van Steenbergen-Weijenburg|first1=Kirsten|last2=de Vroege|first2=L|last3=Ploeger|first3=R.R.|year=2010|title=Validation of the PHQ-9 as a Screening Instrument for Depression in Diabetes Patients in Specialized Outpatient Clinics|journal=BMC Health Services Research|volume=10|pages=1–6|doi=10.1186/1472-6963-10-235|pmc=2927590|pmid=20704720}}</ref> [[w:HIV/AIDS|HIV-AIDS]],<ref>{{Cite journal|last1=Monohan|first1=P.O.|last2=Shacham|first2=E|last3=Reece|first3=M|year=2009|title=Validity/Reliability of PHQ-9 and PHQ-2 Depression Scales Among Adults Living with HIV/AIDS in Western Kenya|journal=Journal of General Internal Medicine|volume=24|issue=2|pages=189–197|doi=10.1007/s11606-008-0846-z|pmc=2629000|pmid=19031037}}</ref> [[w:chronic pain | chronic pain]], [[w:arthritis | arthritis]], [[w:fibromyalgia | fibromyalgia]], [[w:epilepsy | epilepsy]], and substance abuse.<ref name=":3" /> It also is used in studies involving patients with physical disabilities as well as older adults, students, and adolescents.<ref name=":3" /> The PHQ-9 is available in over 30 languages<ref>{{Cite web|url=http://www.apa.org/pi/about/publications/caregivers/practice-settings/assessment/tools/patient-health.aspx|title=Patient Health Questionnaire (PHQ-9 & PHQ-2)|website=American Psychological Association}}</ref> and it has been validated for use in different ethnicities.<ref name=":3" /> Currently Pfizer owns the copyright of the PHQ-9, but allows it to accessed for free.<ref name=":0" /> == Related instruments == The '''PHQ-2''' is a shortened version of the PHQ-9. It contains the first 2 questions of the PHQ-9 and takes less than a minute to administer. A score of 3 or greater on the PHQ-2 will generally lead to the subsequent administration of the PHQ-9. The Veterans Administration uses this method to screen for depression in patients.<ref name=":3" /> The '''PHQ-8''' consists of all of the PHQ-9 instruments except for the last question (suicidal thoughts). It is usually used in research settings in non-depressive patients.<ref>{{Cite web|url=https://phqscreeners.pfizer.edrupalgardens.com/sites/g/files/g10016261/f/201412/instructions.pdf|title=Instructions for Patient Health Questionnaire and GAD-7 Measures|website=Pfizer}}</ref> Researchers generally use the PHQ-8 because timing and resource restraints may leave researchers unable to intervene with study participants that indicate suicidal thoughts. The absence of the ninth question has little effect on scoring between the PHQ-8 and PHQ-9. A study found that scores between the two tests are highly correlated (r=0.998).<ref name=":4" /> The '''PHQ-9 Modified for adolescents''' is a modified version of the original PHQ-9 that was developed to screen for major depressive disorder among adolescents in primary settings. It is commonly called PHQ-A<ref>{{cite journal | last1= López-Torres first1= S. | last2= Pérez-Pedrogo | first2= C. | last3=Sánchez-Cardona |first3= I.| last4= Sánchez-Cesáreo |first4= M. | journal=Current Psychology | title=Psychometric Properties of the PHQ-A among a Sample of Children and Adolescents in Puerto Rico | volume=41 | issue=1 | pages=90–98 | date= January 2022 | issn=1046-1310, 1936-4733 | doi=10.1007/s12144-019-00468-7}}</ref><ref>{{cite journal | last1= Manfro | first1= P. H. | last2= Pereira | First2= R. B.|last3=Rosa |first3= M.|last4=Cogo-Moreira |first4= H.|last5=Fisher| first5= H. L.| last6=Kohrt|first6= B. A.|last7=Mondelli |first7= V.|last8=Kieling|first8= C. | journal=European Child & Adolescent Psychiatry | title=Adolescent depression beyond DSM definition: a network analysis | volume=32 | issue=5 | pages=881–892 | date= May 2023 | issn=1018-8827, 1435-165X | doi=10.1007/s00787-021-01908-1}}</ref>, although this name originally refers to the whole PRIME-MD PHQ<ref>{{cite journal | last1=Johnson |first1=J. G.| last2 = Harris | first2 = E. S. | last3= Spitzer | first3 = R. L. | last4 =Williams| first4= J. B. W. | journal=Journal of Adolescent Health | title=The patient health questionnaire for adolescents | volume=30 | issue=3 | pages=196–204 | date= March 2002 | issn=1054139X | doi=10.1016/S1054-139X(01)00333-0}}</ref>, a 67-item questionnaire that also covers other disorders such as generalized anxiety disorder, bulimia nervosa and substance use disorders. The '''PHQ-15''' is the 15-question scale from the PHQ that asks about 15 symptoms relating to somatoform disorders. The questions on the PHQ-15 account for 90% of all symptoms that providers observe in the primary care setting<ref name=":4" /> Patients must rate how certain symptoms bothered them over the last month. Responses range from "not at all" (a score of 0) to "bothered a lot" (a score of 2). Higher scores on the PHQ-15 are strongly associated with functional impairment, disability and health care use.<ref name=":4" /> The '''GAD-7''' is a 7-question anxiety screening instrument developed in 2006. Like the PHQ-9, scores range from 0 to 27 with scores of 5, 10, and 15 indicating mild, moderate, and severe anxiety. Unlike the PHQ-9, clinicians use the GAD-7 to assess the severity of anxiety only. A clinical interview must be given to determine the presence and type of anxiety. The GAD-2 is a 2-question shortened version of the GAD-7 that uses the first two questions from the GAD-7. A total score that is greater than 3 indicates that a clinician should administer the full GAD-7 and conduct a clinical interview to assess the presence and type of anxiety disorder.<ref name=":4" /> == References == {{reflist}} == External links == * [https://www.phqscreeners.com/ Official site] {{:{{BASEPAGENAME}}/Navbox}} {{CourseCat}} 6kkttgdf5h01vixyp11cu3dx34dq8h0 0 275343 2691709 2310530 2024-12-12T23:03:14Z Yinuo Liu 2935880 adjust format, add banner + instrument table 2691709 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{short description|Hypomania Checklist}} The '''Hypomania Checklist''' ('''HCL-32''') is a [[questionnaire]] developed by Dr. [[w:Jules Angst|Jules Angst]] to identify [[w:hypomania|hypomanic]] features in patients with [[w:major depressive disorder|major depressive disorder]] in order to help recognize [[w:bipolar disorder|bipolar II disorder]] and other bipolar spectrum disorders<ref name="Bowling A 281–91">{{cite journal |vauthors=Bowling A |title=Mode of questionnaire administration can have serious effects on data quality |journal=Journal of Public Health |volume=27 |issue=3 |pages=281–91 |year=2005 |pmid=15870099 |doi=10.1093/pubmed/fdi031 |url=http://eprints.kingston.ac.uk/17190/ |doi-access=free }}</ref> when people seek help in primary care and other general medical settings. It asks about 32 behaviors and mental states that are either aspects of [[w:hypomania|hypomania]] or features associated with mood disorders. It uses short phrases and simple language, making it easy to read. The [[w:University of Zurich|University of Zurich]] holds the copyright, and the HCL-32 is available for use at no charge. More recent work has focused on validating translations and testing whether shorter versions still perform well enough to be helpful clinically.<ref name=":1">{{cite journal |vauthors=Forty L, Kelly M, Jones L, Jones I, Barnes E, Caesar S, Fraser C, Gordon-Smith K, Griffiths E, Craddock N, Smith DJ |title=Reducing the Hypomania Checklist (HCL-32) to a 16-item version |journal=Journal of Affective Disorders |volume=124 |issue=3 |pages=351–6 |year=2010 |pmid=20129673 |doi=10.1016/j.jad.2010.01.004 }}</ref> Recent meta-analyses find that it is one of the most accurate assessments available for detecting hypomania, doing better than other options at recognizing bipolar II disorder.<ref>{{Cite journal|last=Carvalho|first=André F.|last2=Takwoingi|first2=Yemisi|last3=Sales|first3=Paulo Marcelo G.|last4=Soczynska|first4=Joanna K.|last5=Köhler|first5=Cristiano A.|last6=Freitas|first6=Thiago H.|last7=Quevedo|first7=João|last8=Hyphantis|first8=Thomas N.|last9=McIntyre|first9=Roger S.|date=2015-02-01|title=Screening for bipolar spectrum disorders: A comprehensive meta-analysis of accuracy studies|journal=Journal of Affective Disorders|language=English|volume=172|pages=337–346|doi=10.1016/j.jad.2014.10.024|issn=0165-0327|pmid=25451435|url=http://www.repositorio.ufc.br/handle/riufc/23654}}</ref><ref>{{Cite journal|last=Takwoingi|first=Yemisi|last2=Riley|first2=Richard D.|last3=Deeks|first3=Jonathan J.|date=2015-11-01|title=Meta-analysis of diagnostic accuracy studies in mental health|url= |journal=Evidence-Based Mental Health|language=en|volume=18|issue=4|pages=103–109|doi=10.1136/eb-2015-102228|issn=1468-960X|pmc=4680179|pmid=26446042}}</ref> ==Development and history== The Hypomania Checklist was built as a more efficient screening measure for hypomania, to be used both in epidemiological research and in clinical use. Existing measures for bipolar disorder focused on identifying personality factors and symptom severity instead of the episodic nature of hypomania or the possible negative consequences in behavioral, affective, or cognitive changes associated.<ref name=":0" /> These measures were mostly used in non-clinical populations to identify individuals at risk and were not used as screening instruments. Initially developed by Jules Angst and Thomas Meyer the HCL-32 is intended to have high sensitivity, allowing clinicians from many countries to diagnose individuals in a clinical population with bipolar disorder, specifically bipolar II disorder.{{cn|date=December 2020}} The questionnaire was translated into English and translated back to German to ensure accuracy. The English version of the HCL has been used as the basis for translation in other languages through the same process. The original study that used the HCL in an Italian and a Swiss sample noted the measure's high sensitivity and a lower sensitivity than other used measures.<ref name=":0" /> The scale includes a checklist of 32 possible symptoms of hypomania, each rated yes or no. The rating "yes" would mean the symptom is present or this trait is "typical of me," and "no" would mean that the symptom is not present or "not typical" for the person.<ref name=":0">{{cite journal |vauthors=Angst J, Adolfsson R, Benazzi F, Gamma A, Hantouche E, Meyer TD, Skeppar P, Vieta E, Scott J |title=The HCL-32: towards a self-assessment tool for hypomanic symptoms in outpatients |journal=Journal of Affective Disorders |volume=88 |issue=2 |pages=217–33 |year=2005 |pmid=16125784 |doi=10.1016/j.jad.2005.05.011 }}</ref> == Short Form Versions == Similar [[w:Reliability (psychometrics)|reliability]] scores were found when only using 16 item assessments versus the traditional 32-item format of the HCL-32.<ref name=":1" /> A score of at least 8 items was found valid and reliable for distinguishing Bipolar Disorder and Major Depressive Disorder{{citation needed|date=October 2015}}. However, the 16-item HCL has not been tested as a standalone section in a hospital setting.<ref name=":0" /> ==Limitations== The HCL suffers from the same problems as other [[w:Self-report inventories|self-report inventories]], in that scores can be easily exaggerated or minimized by the person completing them. Like all questionnaires, the way the instrument is administered can influence the final score. If a patient is asked to fill out the form in front of other people in a clinical environment, for instance, social expectations may elicit a different response compared to administration via a postal survey.<ref name="Bowling A 281–91"/> That said, the online version of the HCL has been shown to be as reliable as the paper version.<ref name="AngstWeb" /> In a study, 73% of patients who completed the HCL-32-R1 were true bipolar cases identified as potential bipolar cases. However, while the HCL-32 is a sensitive instrument for hypomanic symptoms, the HCL-32-R1 does not accurately differentiate between Bipolar I and Bipolar II.<ref name=":0" /><ref name=AngstWeb>{{cite web|last1=Angst|date=June 2007|title=Hypomania Check List (HCL-32 R1) Manual|url=http://www.lakarhuset.com/docs/HCL_32_R1_Manual.pdf|accessdate=23 November 2015}}</ref> The HCL-32 has not been compared with other commonly used screening tools for bipolar disorder, such as the [[Young Mania Rating Scale]] and the [[w:Parent General Behavior Inventory|General Behavior Inventory]]. == See also == * [[w:Bipolar disorder|Bipolar disorder]] * [[w:Bipolar disorder in children|Bipolar disorder in children]] == Further reading == * {{cite journal |last1=Birmaher |first1=Boris |last2=Brent |first2=David |author3=AACAP Work Group on Quality Issues |date=November 2007 |title=Practice Parameter for the Assessment and Treatment of Children and Adolescents With Depressive Disorders |journal=Journal of the American Academy of Child and Adolescent Psychiatry |volume=46 |issue=11 |pages=1503–1526 |doi=10.1097/chi.0b013e318145ae1c |doi-access=free |pmid=18049300 |url=http://www.jaacap.com/article/S0890-8567(09)62053-0/fulltext }} * {{cite journal |last1=McClellan |first1=Jon |last2=Kowatch |first2=Robert |last3=Findling |first3=Robert L. |author4=Work Group on Quality Issues |date=January 2007 |title=Practice parameter for the assessment and treatment of children and adolescents with bipolar disorder |journal=Journal of the American Academy of Child and Adolescent Psychiatry |volume=46 |issue=1 |pages=107–25 |doi=10.1097/01.chi.0000242240.69678.c4 |doi-access=free |pmid=17195735 |url=http://www.jaacap.com/article/S0890-8567(09)61968-7/fulltext }} == External links == {{Wikiversity|Hypomania Checklist}} * Official Manual and [http://www.lakarhuset.com/docs/HCL_32_R1_Manual.pdf Question Breakdown] {{Mental and behavioural disorders}} == References == {{Reflist}} {{:{{BASEPAGENAME}}/Navbox}} [[Category:Mental and behavioral disorders]] [[Category:Psychological testing]] [[Category:Psychiatric instruments: child and adolescent psychiatry]] [[Category:Psychiatric instruments: mania]] e45u4gwsrdfepkxtcd7ug0iv682zgao 2691710 2691709 2024-12-12T23:03:53Z Yinuo Liu 2935880 debug template errors 2691710 wikitext text/x-wiki <noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude> {{short description|Hypomania Checklist}} The '''Hypomania Checklist''' ('''HCL-32''') is a [[questionnaire]] developed by Dr. [[w:Jules Angst|Jules Angst]] to identify [[w:hypomania|hypomanic]] features in patients with [[w:major depressive disorder|major depressive disorder]] in order to help recognize [[w:bipolar disorder|bipolar II disorder]] and other bipolar spectrum disorders<ref name="Bowling A 281–91">{{cite journal |vauthors=Bowling A |title=Mode of questionnaire administration can have serious effects on data quality |journal=Journal of Public Health |volume=27 |issue=3 |pages=281–91 |year=2005 |pmid=15870099 |doi=10.1093/pubmed/fdi031 |url=http://eprints.kingston.ac.uk/17190/ |doi-access=free }}</ref> when people seek help in primary care and other general medical settings. It asks about 32 behaviors and mental states that are either aspects of [[w:hypomania|hypomania]] or features associated with mood disorders. It uses short phrases and simple language, making it easy to read. The [[w:University of Zurich|University of Zurich]] holds the copyright, and the HCL-32 is available for use at no charge. More recent work has focused on validating translations and testing whether shorter versions still perform well enough to be helpful clinically.<ref name=":1">{{cite journal |vauthors=Forty L, Kelly M, Jones L, Jones I, Barnes E, Caesar S, Fraser C, Gordon-Smith K, Griffiths E, Craddock N, Smith DJ |title=Reducing the Hypomania Checklist (HCL-32) to a 16-item version |journal=Journal of Affective Disorders |volume=124 |issue=3 |pages=351–6 |year=2010 |pmid=20129673 |doi=10.1016/j.jad.2010.01.004 }}</ref> Recent meta-analyses find that it is one of the most accurate assessments available for detecting hypomania, doing better than other options at recognizing bipolar II disorder.<ref>{{Cite journal|last=Carvalho|first=André F.|last2=Takwoingi|first2=Yemisi|last3=Sales|first3=Paulo Marcelo G.|last4=Soczynska|first4=Joanna K.|last5=Köhler|first5=Cristiano A.|last6=Freitas|first6=Thiago H.|last7=Quevedo|first7=João|last8=Hyphantis|first8=Thomas N.|last9=McIntyre|first9=Roger S.|date=2015-02-01|title=Screening for bipolar spectrum disorders: A comprehensive meta-analysis of accuracy studies|journal=Journal of Affective Disorders|language=English|volume=172|pages=337–346|doi=10.1016/j.jad.2014.10.024|issn=0165-0327|pmid=25451435|url=http://www.repositorio.ufc.br/handle/riufc/23654}}</ref><ref>{{Cite journal|last=Takwoingi|first=Yemisi|last2=Riley|first2=Richard D.|last3=Deeks|first3=Jonathan J.|date=2015-11-01|title=Meta-analysis of diagnostic accuracy studies in mental health|url= |journal=Evidence-Based Mental Health|language=en|volume=18|issue=4|pages=103–109|doi=10.1136/eb-2015-102228|issn=1468-960X|pmc=4680179|pmid=26446042}}</ref> ==Development and history== The Hypomania Checklist was built as a more efficient screening measure for hypomania, to be used both in epidemiological research and in clinical use. Existing measures for bipolar disorder focused on identifying personality factors and symptom severity instead of the episodic nature of hypomania or the possible negative consequences in behavioral, affective, or cognitive changes associated.<ref name=":0" /> These measures were mostly used in non-clinical populations to identify individuals at risk and were not used as screening instruments. Initially developed by Jules Angst and Thomas Meyer the HCL-32 is intended to have high sensitivity, allowing clinicians from many countries to diagnose individuals in a clinical population with bipolar disorder, specifically bipolar II disorder.{{cn|date=December 2020}} The questionnaire was translated into English and translated back to German to ensure accuracy. The English version of the HCL has been used as the basis for translation in other languages through the same process. The original study that used the HCL in an Italian and a Swiss sample noted the measure's high sensitivity and a lower sensitivity than other used measures.<ref name=":0" /> The scale includes a checklist of 32 possible symptoms of hypomania, each rated yes or no. The rating "yes" would mean the symptom is present or this trait is "typical of me," and "no" would mean that the symptom is not present or "not typical" for the person.<ref name=":0">{{cite journal |vauthors=Angst J, Adolfsson R, Benazzi F, Gamma A, Hantouche E, Meyer TD, Skeppar P, Vieta E, Scott J |title=The HCL-32: towards a self-assessment tool for hypomanic symptoms in outpatients |journal=Journal of Affective Disorders |volume=88 |issue=2 |pages=217–33 |year=2005 |pmid=16125784 |doi=10.1016/j.jad.2005.05.011 }}</ref> == Short Form Versions == Similar [[w:Reliability (psychometrics)|reliability]] scores were found when only using 16 item assessments versus the traditional 32-item format of the HCL-32.<ref name=":1" /> A score of at least 8 items was found valid and reliable for distinguishing Bipolar Disorder and Major Depressive Disorder{{citation needed|date=October 2015}}. However, the 16-item HCL has not been tested as a standalone section in a hospital setting.<ref name=":0" /> ==Limitations== The HCL suffers from the same problems as other [[w:Self-report inventories|self-report inventories]], in that scores can be easily exaggerated or minimized by the person completing them. Like all questionnaires, the way the instrument is administered can influence the final score. If a patient is asked to fill out the form in front of other people in a clinical environment, for instance, social expectations may elicit a different response compared to administration via a postal survey.<ref name="Bowling A 281–91"/> That said, the online version of the HCL has been shown to be as reliable as the paper version.<ref name="AngstWeb" /> In a study, 73% of patients who completed the HCL-32-R1 were true bipolar cases identified as potential bipolar cases. However, while the HCL-32 is a sensitive instrument for hypomanic symptoms, the HCL-32-R1 does not accurately differentiate between Bipolar I and Bipolar II.<ref name=":0" /><ref name=AngstWeb>{{cite web|last1=Angst|date=June 2007|title=Hypomania Check List (HCL-32 R1) Manual|url=http://www.lakarhuset.com/docs/HCL_32_R1_Manual.pdf|accessdate=23 November 2015}}</ref> The HCL-32 has not been compared with other commonly used screening tools for bipolar disorder, such as the [[Young Mania Rating Scale]] and the [[w:Parent General Behavior Inventory|General Behavior Inventory]]. == See also == * [[w:Bipolar disorder|Bipolar disorder]] * [[w:Bipolar disorder in children|Bipolar disorder in children]] == Further reading == * {{cite journal |last1=Birmaher |first1=Boris |last2=Brent |first2=David |author3=AACAP Work Group on Quality Issues |date=November 2007 |title=Practice Parameter for the Assessment and Treatment of Children and Adolescents With Depressive Disorders |journal=Journal of the American Academy of Child and Adolescent Psychiatry |volume=46 |issue=11 |pages=1503–1526 |doi=10.1097/chi.0b013e318145ae1c |doi-access=free |pmid=18049300 |url=http://www.jaacap.com/article/S0890-8567(09)62053-0/fulltext }} * {{cite journal |last1=McClellan |first1=Jon |last2=Kowatch |first2=Robert |last3=Findling |first3=Robert L. |author4=Work Group on Quality Issues |date=January 2007 |title=Practice parameter for the assessment and treatment of children and adolescents with bipolar disorder |journal=Journal of the American Academy of Child and Adolescent Psychiatry |volume=46 |issue=1 |pages=107–25 |doi=10.1097/01.chi.0000242240.69678.c4 |doi-access=free |pmid=17195735 |url=http://www.jaacap.com/article/S0890-8567(09)61968-7/fulltext }} == External links == {{Wikiversity|Hypomania Checklist}} * Official Manual and [http://www.lakarhuset.com/docs/HCL_32_R1_Manual.pdf Question Breakdown] == References == {{Reflist}} {{:{{BASEPAGENAME}}/Navbox}} [[Category:Mental and behavioral disorders]] [[Category:Psychological testing]] [[Category:Psychiatric instruments: child and adolescent psychiatry]] [[Category:Psychiatric instruments: mania]] p9tfzgmc2k7x4efvtsomg8q8f0vtfcn 0 283011 2691700 2628780 2024-12-12T22:27:08Z Dc.samizdat 2856930 2691700 wikitext text/x-wiki {{Polyscheme|radius=a '''commentary''' on|active=is recently published research which may not have received peer review}} [[Category:Geometry]] [[Category:Polyscheme]] qafkb03x6n7lhbo70zv2pypys4dclnp 0 285380 2691626 2691404 2024-12-12T13:45:59Z Young1lim 21186 /* Applications */ 2691626 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241212.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> jzva7ny7jvz5a2v24m609g5j49qeaij 0 285384 2691711 2691522 2024-12-12T23:04:31Z Young1lim 21186 /* Linking Libraries */ 2691711 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241209.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241213.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] bhzzft0n9khfxilnhe3qy3u921ouzh0 2 289273 2691702 2686740 2024-12-12T22:31:36Z Dc.samizdat 2856930 /* Symmetries */ 2691702 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - November 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four or more orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere with these objects distributed on its surface; the black hole at its center is the 4-ball of empty space it surrounds. The observable universe of galaxies may be visualized as a 4-sphere expanding radially from an origin point at velocity <math>c</math>, the propagation speed of light through 3-space, which is also the invariant velocity of all mass-carrying objects through 4-space. The propagation speed of light through 4-space <math>c_4</math> is actually <math>c < c_4 < 2c</math>. This view is compatible with the theories of special and general relativity, and with the quantum mechanical atomic theory. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice. Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} {{Reflist|40em}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} tbgkgtc777qkyhyv01cnv6nmfk0qf8y 0 295784 2691611 2634412 2024-12-12T12:10:13Z Lbeaumont 278565 /* Wise Practices */ Added "Give care" 2691611 wikitext text/x-wiki '''It is [[Wisdom|wise]] to allow more people to [[Evolving Governments/Good Government#/media/File:Evaluating Good Government.jpg|meet more of their needs]].''' Resources to help you live more wisely are assembled here for your use. [[File:Vision_of_the_Future.jpg|thumb|270x270px|We can wisely create our future!]] {{TOC right | limit|limit=2}} == For Everyone: == === Freely Available Learning Resources === These on-line courses are freely available worldwide. * [[The Wise Path]] provides a guide to learning resources and practices that can enable you to live more wisely. * The [[Living Wisely]] curriculum is a portal into many wisdom-related course materials. * The [[Wisdom/Curriculum|Applied Wisdom]] curriculum provides direct access to many wisdom-related learning resources. === Relevant Blogs === These blogs provide wise advice and insights. * [https://lelandbeaumont.substack.com/ Seeking Real Good] * [https://lelandbeaumont.substack.com/p/the-future-of-education-is-learning The Future of Education is Learning]. === Reading Lists === Reading these books can help you become wiser. * [https://www.librarything.com/list/10724/all/Wisdom Wisdom] – a list of books to help you progress toward wisdom. * [https://www.librarything.com/list/44495/all/How-it-can-be How can it be] – a list of books that describe possibilities for a wiser future. * [https://www.librarything.com/list/10041/all/Attaining-Belief Attaining Belief] – a list of books describing how our beliefs originate and are shaped over time. * [https://www.librarything.com/list/20350/all/Creating-Possibilities Creating Possibilities] – a list of books that help us [[Solving Problems|solve problems]], clarify and reframe problems, create alternative solutions, think creativity, and choose a better path forward. * [https://www.librarything.com/list/10594/all/Secular-Ethics Secular Ethics] – a list of books that help you decide the right thing to do. * [https://www.librarything.com/list/1167/all/Rethinking-Money Rethinking Money] – books on this list highlight problems with today's dominant money systems or suggest alternatives to those systems. === Videos === * [https://www.youtube.com/watch?v=0N5qj5Ck7wk&t=3s Advancing human rights], worldwide * [https://www.youtube.com/watch?v=evrPk6tCETw&t=2s Seeking Real Good] * [https://youtu.be/Xg9iHUkYjGY The Wise Path] * [https://www.youtube.com/watch?v=xrXLVoTf7Kk Reimagining Humanity] === Wise Practices === Actions express [[wisdom]]. We invite you to attain these skills required to practice wisdom, and live wisely. [[File:Origins_and_progression_of_wisdom.webp|thumb|Origins and Progression of Wisdom]] # [[Living Wisely/Take Care|Take care]]. Give care # Uphold [[Living Wisely#Assignment|the four agreements]]. # [[Living Wisely/advance no falsehoods|Advance no falsehoods]]. # [[Knowing How You Know|Know how you know]]. # [[Facing Facts/Reality is our common ground|Embrace reality]] and [[Facing Facts|face facts]]. # [[Seeking True Beliefs|Seek true beliefs]]. Insist on [[Intellectual Honesty|intellectual honesty]]. # [[Practicing Dialogue|Practice dialogue]] and [[candor]]. # Become [[Studying Emotional Competency|emotionally competent]]. # Live the [[Virtues|moral virtues]]. # Respect [[dignity]] and [[Assessing Human Rights|preserve human rights]], worldwide. # [[Living the Golden Rule|Live the Golden Rule]]. # Clarify your [[Moral Reasoning|moral reasoning]]. # Adopt a [[Global Perspective|global perspective]]. # Focus on [[What Matters|what matters]]. # Undertake the [[Grand Challenges|grand challenges]]. # [[Doing Good|Do good]]. # Enjoy [[Living Wisely/Seeking Real Good|seeking ''real'' good]] throughout your life. # Choose to [[Living Wisely|live wisely]]. # Find [[Finding Common Ground|common ground]]. # [[Coming Together|Come together]]. Many people find that regular practices such as [[Meditation|meditating]] or [[w:Diary|journaling]] help them live more wisely. === Projects === We can apply wise practices to several [[Living Wisely/Seeking Real Good|real good]] projects and help to transform our world, now and into the future. # [https://www.youtube.com/watch?v=0N5qj5Ck7wk&t=7s Advancing human rights, worldwide] may be the most effective action we can take to address many of the world’s [[Grand challenges|greatest challenges]]. These include war, refugee displacements, immigration issues, oppression, torture, jehad, terrorism, poverty, access to education, systemic inequality, endemic diseases, and many more. We can progress [[Assessing Human Rights/Beyond Olympic Gold|beyond Olympic gold]]. We can [https://thefulcrum.us/advancing-human-rights-worldwide advance human rights, worldwide]. # We can directly address the [[grand challenges]], the greatest, most pervasive and persistent problems facing humanity that also offer the most promising opportunities. We can choose to make [https://thefulcrum.us/prioritizing-the-grand-challenges addressing the grand challenges our priority]. # By [[Evolving Governments|evolving governments]] we can become more agile and better meet human needs, worldwide. # We can work to ensure [[Level 5 Research Center|the future that emerges]] embraces [[Level 5 Research Center#Values|pro-social values]]. == For Academics == For many years philosopher [[w:Nicholas_Maxwell|Nicholas Maxwell]] has advocated a specific plan to transform academia from ''knowledge inquiry'' to ''wisdom inquiry''. His approach is described by the following materials. * [https://philpapers.org/rec/MAXCUS Can Universities Save Us From Disaster?] * [https://www.ucl.ac.uk/from-knowledge-to-wisdom/whatneedstochange What Needs to Change] * [https://www.frontiersin.org/articles/10.3389/frsus.2021.631631/full How Universities Have Betrayed Reason and Humanity—And What's to Be Done About It] * ''[https://philarchive.org/rec/MAXFKT-2 From Knowledge to Wisdom]'' The [https://pathes.org Philosophy and Theory of Higher Education Society] provides a collaborative space for scholars to come together, in reflecting on the values of the university as an institution and on higher education as educational practices. === Philosophy === [[w:Philosophy|Philosophy]] is literally “love of wisdom”. More practically, philosophy is what happens when we begin to think for ourselves. These resources provide in-depth treatments of many philosophical issues and can help us live more wisely. * The [[w:Stanford_Encyclopedia_of_Philosophy|Stanford Encyclopedia of Philosophy]] (SEP) combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. * The [[w:Internet_Encyclopedia_of_Philosophy|Internet Encyclopedia of Philosophy]] (IEP) is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. * [[w:RationalWiki|RationalWiki]] is an online wiki which is written from a scientific skeptic, secular, and progressive perspective. Its stated goals are to “analyze and refute pseudoscience and the anti-science movement, document crank ideas, explore conspiracy theories, authoritarianism, and fundamentalism, and analyze how these subjects are handled in the media.” * [[w:PhilPapers|PhilPapers]] is an interactive academic database of journal articles in philosophy. * [https://philpeople.org/ PhilPeople] is an online directory of philosophers, a social network for philosophers, and a tool for keeping up with the philosophical profession. * A [[Philosophy|philosophy curriculum]] is emerging on Wikiversity. You may wish to help [[Creating Wikiversity Courses|develop those courses]]. == For Researchers == Explore the frontiers of wisdom. * The ''[[Wisdom Research|Wisdom and the Future Research Center]]'' is where researchers are exploring the question '''How can we wisely create our future?''' * The ''[[Level 5 Research Center]]'' is where researchers are exploring the question '''How can we best shape the emergence of Level 5?''' [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] jphmohc698xkz01mv4qncrsvtr7an7s 0 299451 2691706 2581226 2024-12-12T22:52:00Z Yinuo Liu 2935880 added instrument table 2691706 wikitext text/x-wiki The '''Bipolar Spectrum Diagnostic Scale''' ('''BSDS''') is a [[Psychiatry|psychiatric]] [[w:Self rating scale|self-rating scale]] created by Ronald Pies in screening for [[w:bipolar disorder|bipolar disorder]] (BD).<ref name=":0">{{Cite journal |last=Youngstrom |first=Eric A. |last2=Egerton |first2=Gregory A. |last3=Genzlinger |first3=Jacquelynne |last4=Freeman |first4=Lindsey K. |last5=Rizvi |first5=Sabeen H. |last6=Van Meter |first6=Anna |date=2018 |title=Improving the global identification of bipolar spectrum disorders: Meta-analysis of the diagnostic accuracy of checklists. |url=http://doi.apa.org/getdoi.cfm?doi=10.1037/bul0000137 |journal=Psychological Bulletin |language=en |volume=144 |issue=3 |pages=315–342 |doi=10.1037/bul0000137 |issn=1939-1455}}</ref> Its initial version consists of a descriptive narrative aimed to capture the nuances and milder variants of BD.<ref name=":1">{{Cite journal |last=Nassir Ghaemi |first=S. |last2=Miller |first2=Christopher J. |last3=Berv |first3=Douglas A. |last4=Klugman |first4=Jeffry |last5=Rosenquist |first5=Klara J. |last6=Pies |first6=Ronald W. |date=2005 |title=Sensitivity and specificity of a new bipolar spectrum diagnostic scale |url=https://linkinghub.elsevier.com/retrieve/pii/S0165032703001964 |journal=Journal of Affective Disorders |language=en |volume=84 |issue=2-3 |pages=273–277 |doi=10.1016/S0165-0327(03)00196-4}}</ref> Upon revision by [[w:Nassir Ghaemi|Nassir Ghaemi]] and colleagues, the scale was developed into two sections for a total of 20 questions.<ref name=":1" /> The BSDS is widely accepted as an important measure of bipolar disorder alongside other [[w:Diagnostic tool|diagnostic tools]] such as the [[Mood Disorder Questionnaire]] and the Bipolar Depression Rating Scale.<ref>{{Cite journal |last=Lee |first=Dongyun |last2=Cha |first2=Boseok |last3=Park |first3=Chul-Soo |last4=Kim |first4=Bong-Jo |last5=Lee |first5=Cheol-Soon |last6=Lee |first6=Sojin |date=2013 |title=Usefulness of the combined application of the Mood Disorder Questionnaire and Bipolar Spectrum Diagnostic Scale in screening for bipolar disorder |url=https://linkinghub.elsevier.com/retrieve/pii/S0010440X12002210 |journal=Comprehensive Psychiatry |language=en |volume=54 |issue=4 |pages=334–340 |doi=10.1016/j.comppsych.2012.10.002}}</ref> == Background == [[w:Bipolar disorder|Bipolar Disorder (BD)]] is a [[w:psychiatric disorder|psychiatric disorder]] defined by intermittent episodes of [[w:Depression (mood)|depression]] and [[w:Hypomania|(hypo)]][[w:mania|mania]] during the individual's lifetime. The [[w:DSM-5|DSM-5]] and [[w:ICD-11|ICD-11]] recognise bipolar disorder as a [[w:spectrum|spectrum]] with three specific subtypes: [[w:bipolar I disorder|Bipolar I disorder]], [[w:Bipolar II disorder|Bipolar II disorder]] and [[w:Cyclothymic Disorder|cyclothymic disorder]]. The [[w:Prevalence|lifetime prevalence]] of BD is approximately 1% in the general population,<ref>{{Cite journal |last=Rowland |first=Tobias A. |last2=Marwaha |first2=Steven |date=2018 |title=Epidemiology and risk factors for bipolar disorder |url=http://journals.sagepub.com/doi/10.1177/2045125318769235 |journal=Therapeutic Advances in Psychopharmacology |language=en |volume=8 |issue=9 |pages=251–269 |doi=10.1177/2045125318769235 |issn=2045-1253 |pmc=6116765 |pmid=30181867}}</ref> but rises to 4% when given the broader definition of bipolar spectrum disorder. As a result of the broad and complex nature of bipolar disorder, [[w:misdiagnosis|misdiagnosis]] is fairly common: 69% of confirmed cases are found to be initially misdiagnosed and more than a third of individuals are misdiagnosed for ten years onwards.<ref>{{Cite web |last=Mooney |first=Brenda |date=2022-05-29 |title=Bipolar Disorder Often Misdiagnosed as Major Depression, Leading to Improper Treatment |url=https://www.usmedicine.com/2022-compendium-of-federal-medicine/bipolar-disorder-often-misdiagnosed-as-major-depression-leading-to-improper-treatment/ |access-date=2023-04-12 |website=U.S. Medicine |language=en-US}}</ref> For individuals with milder symptoms of BD, this seems to be even more prevalent. The BSDS was devised to estimate not only severe cases of bipolar disorder, but also milder variants in a more sensitive manner. The scale is ideal for screening, but not for diagnosing BD as the 19 questions do not accurately reflect the main criterion of the [[w:DSM-5|DSM-5]].<ref name=":2">{{Cite journal |last=Harrison |first=Paul J. |last2=Geddes |first2=John R. |last3=Tunbridge |first3=Elizabeth M. |date=2018 |title=The Emerging Neurobiology of Bipolar Disorder |url=https://linkinghub.elsevier.com/retrieve/pii/S0166223617302126 |journal=Trends in Neurosciences |language=en |volume=41 |issue=1 |pages=18–30 |doi=10.1016/j.tins.2017.10.006 |pmc=5755726 |pmid=29169634}}</ref> The scale has however been found to accurately rule out a diagnosis of BD altogether for an individual.<ref name=":2" /> == Development == The original English Version of the BSDS consists of a descriptive passage with nineteen statements ending with a blank space. Patients are first advised to read through the entire passage before starting the assessment. Once completed, they are asked to place a check next to each of the nineteen items they feel relates to their personal experience of BD.<ref name=":3">{{Cite journal |last=Zaratiegui |first=Rodolfo M. |last2=Vázquez |first2=Gustavo H. |last3=Lorenzo |first3=Laura S. |last4=Marinelli |first4=Marcia |last5=Aguayo |first5=Silvia |last6=Strejilevich |first6=Sergio A. |last7=Padilla |first7=Eduardo |last8=Goldchluk |first8=Aníbal |last9=Herbst |first9=Luis |last10=Vilapriño |first10=Juan J. |last11=Bonetto |first11=Gerardo García |last12=Cetkovich-Bakmas |first12=Marcelo G. |last13=Abraham |first13=Estela |last14=Kahn |first14=Clara |last15=Whitham |first15=Elizabeth A. |date=2011 |title=Sensitivity and specificity of the mood disorder questionnaire and the bipolar spectrum diagnostic scale in Argentinean patients with mood disorders |url=https://linkinghub.elsevier.com/retrieve/pii/S0165032711000978 |journal=Journal of Affective Disorders |language=en |volume=132 |issue=3 |pages=445–449 |doi=10.1016/j.jad.2011.03.014}}</ref> Each check is worth one point. The passage is written entirely in a [[w:Third-person narrative|third person narrative]]. When assessed by [[w:Nassir Ghaemi|Nassir Ghaemi]] and colleagues, the original scale demonstrated a high diagnostic sensitivity at 0.76, meaning that most people with clinicians' DSM-5-based cases were accurately diagnosed.<ref name=":3" /> The BSDS also correctly identified 85% of unipolar-depressed patients as not having bipolar disorder despite similarities in symptoms, indicating a high specificity score.<ref name=":3" /> To improve the original version, Ghaemi created an additional section for the BSDS. This section involved a 4-item [[Likert scale]] assessing the extent to which individuals felt that the passage related to their own experience of BD. The 4 item scale includes statements of "This story fits me very well." (worth 6 points), "This story fits me fairly well." (worth 4 points), "This story fits me to some degree but not in most respects." (worth 2 points), to "This story does not really describe me at all." (worth 0 points).<ref name=":3" /><ref name=":2" /> The abridged version of BSDS scores range from 0-25 points with the positive threshold for diagnosis at 13 points and above. The likelihood of BD according to the BSDS is given based on the overall score of both sections.<ref name=":0" /><ref name=":3" /> Scores of 0-6 indicates a "highly unlikely" chance of having BD, 7-12 indicates a "low probability", 13-19 indicates a "moderate probability", and a score of 20-25 indicates a "high probability". Ghaemi's BSDS version increased specificity from the original version from 0.85 to 0.93. The BSDS has since been adjusted and adapted for several other global populations, including Persia, Turkey, and Mexico.<ref>{{Cite journal |last=Zimmerman |first=Mark |last2=Galione |first2=Janine N |last3=Chelminski |first3=Iwona |last4=Young |first4=Diane |last5=Ruggero |first5=Camilo J |date=2010-08-16 |title=Performance of the Bipolar Spectrum Diagnostic Scale in psychiatric outpatients: Performance of the Bipolar Spectrum Diagnostic Scale |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1399-5618.2010.00840.x |journal=Bipolar Disorders |language=en |volume=12 |issue=5 |pages=528–538 |doi=10.1111/j.1399-5618.2010.00840.x}}</ref><ref>{{Cite journal |last=İnce |first=Bahri |last2=Cansız |first2=Alparslan |last3=Ulusoy |first3=Sevinç |last4=Yavuz |first4=Kasım Fatih |last5=Kurt |first5=Erhan |last6=Altınbaş |first6=Kürşat |date=2019 |title=Reliability and Validity Study of the Turkish Version of Bipolar Spectrum Diagnostic Scale |url=https://pubmed.ncbi.nlm.nih.gov/32594489/ |journal=Turk Psikiyatri Dergisi = Turkish Journal of Psychiatry |volume=30 |issue=4 |pages=272–278 |issn=2651-3463 |pmid=32594489}}</ref><ref>{{Cite journal |last=Sánchez de la Cruz |first=Juan Pablo |last2=Fresán |first2=Ana |last3=González Moralez |first3=Diana Laura |last4=López-Narváez |first4=María Lilia |last5=Tovilla-Zarate |first5=Carlos Alfonso |last6=Pool-García |first6=Sherezada |last7=Juárez-Rojop |first7=Isela |last8=Hernández-Díaz |first8=Yazmín |last9=González-Castro |first9=Thelma Beatriz |last10=Vera-Campos |first10=María de Lourdes |last11=Velázquez-Sánchez |first11=Patricia |date=2018-11-27 |title=Validation of the Bipolar Spectrum Diagnostic Scale in Mexican Psychiatric Patients |url=https://pubmed.ncbi.nlm.nih.gov/30477597/ |journal=The Spanish Journal of Psychology |volume=21 |pages=E60 |doi=10.1017/sjp.2018.59 |issn=1988-2904 |pmid=30477597}}</ref> == Reliability and validity == The BSDS is a well validated diagnostic tool with a high [[w:Sensitivity and specificity|sensitivity]] (0.76) and [[w:Sensitivity and specificity|specificity]] (0.93) score.<ref name=":1" /> It was also found to have a high [[w:Negative Predictive Value|Negative Predictive Value (NPV)]] of 0.87, suggesting that 87% of the patients who scored below 13 points on the BSDS were correctly identified as not having BD. However, the BSDS was found to have a low [[w:Positive Predictive Value|Positive Predictive Value (PPV)]] of 0.36.<ref>{{Cite journal |last=Shabani |first=Amir |last2=Mirzaei Khoshalani |first2=Mosleh |last3=Mahdavi |first3=Seyedreza |last4=Ahmadzad-Asl |first4=Masoud |date=2019 |title=Screening bipolar disorders in a general hospital: Psychometric findings for the Persian version of mood disorder questionnaire and bipolar spectrum diagnostic scale |url=https://pubmed.ncbi.nlm.nih.gov/31456972 |journal=Medical Journal of the Islamic Republic of Iran |volume=33 |pages=48 |doi=10.34171/mjiri.33.48 |issn=1016-1430 |pmc=6708087 |pmid=31456972}}</ref> Zimmermann et al found a NPV as high as 0.98 and a low PPV of 0.16 when using a representative sample size of 1100 outpatients.<ref>{{Cite journal |last=Zimmerman |first=Mark |last2=Galione |first2=Janine N. |date=2011-09-20 |title=Screening for Bipolar Disorder with the Mood Disorders Questionnaire: A Review |url=https://journals.lww.com/00023727-201109200-00001 |journal=Harvard Review of Psychiatry |language=en |volume=19 |issue=5 |pages=219–228 |doi=10.3109/10673229.2011.614101 |issn=1067-3229}}</ref> This PPV score demonstrates a vulnerability to overdiagnosing BD. In a [[w:systematic review|systematic review]] and [[meta-analysis]] investigating the accuracy of self-report scales for detecting Bipolar Disorder, the BSDS was found to be one of the best performing options along with the Mood Disorder Questionnaire.<ref>{{Cite journal |last=Sayyah |first=Mehdi |last2=Delirrooyfard |first2=Ali |last3=Rahim |first3=Fakher |date=2022 |title=Assessment of the diagnostic performance of two new tools versus routine screening instruments for bipolar disorder: a meta-analysis |url=http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1516-44462022000300349&tlng=en |journal=Brazilian Journal of Psychiatry |volume=44 |issue=3 |pages=349–361 |doi=10.1590/1516-4446-2021-2334 |issn=1809-452X |pmc=9169473 |pmid=35588536}}</ref> The BSDS may do better than other scales at detecting different subtypes of bipolar disorder which do not involve a full manic episode, such as bipolar II or cyclothymic disorder. ==Translations== The following table showcases the translations available for the BSDS {| class="wikitable" |+ Translations Table |- ! Language !! Source Article !! BSDS PDF !! Peer Review !! Psychometrics |- | Arabic || N/A || [https://drive.google.com/drive/folders/15R8Q2b6peKPLzOGXklEgpz97ya7D1z-A Arabic with English] || No || |- | Chinese || [https://onlinelibrary.wiley.com/doi/full/10.1111/j.1365-2702.2010.03390.x Chu et al., 2010] || || Yes || |- | English || [https://www.sciencedirect.com/science/article/pii/S0165032703001964 Ghaemi et al., 2005] || [https://drive.google.com/drive/folders/1W-Sf9HJcvsdP0uNj1aduPi-C3r11kEhs English Version] || Yes || |- | Hindi || || || Yes || |- | Italian || || || || |- | Japanese || || || Yes || |- | Korean || [https://www.koreamed.org/SearchBasic.php?RID=2341496 Wang et al., 2008] || [https://drive.google.com/drive/folders/1J74HxvlrPdGkYkhg3ChIgGQ3gllqovn3 Korean Version] || Yes || |- | Persian || [https://pubmed.ncbi.nlm.nih.gov/19111028/#:~:text=Conclusion%3A%20The%20Persian%20Bipolar%20Spectrum,more%20effective%20than%20either%20alone. Shabani et al., 2009] || || Yes || |- | Portuguese || [https://www.scielo.br/j/jbpsiq/a/KPvWKbGN8N5sQ4Xc6CFS8HJ#:~:text=The%20Brazilian%20Portuguese%20version%20of%20the%20BSDS%20is%20a%20brief,used%20to%20screen%20psychiatric%20disorders. Castelo et al., 2010] || [https://drive.google.com/drive/folders/1IfMt9lTqp1yILEy6fVR74U1risTyrBGF Portuguese Version] || Yes || |- | Spanish || [https://drive.google.com/drive/folders/17WQPP2U_OoQQykU6avS2BNx7-ueAbQnj Vazquez et al., 2010] || [https://drive.google.com/drive/folders/1mpgTORKv8o5yrPh3XK6RitpNavyfZV9K Spanish Version] || Yes (need to check) || |- | Turkish || [https://pubmed.ncbi.nlm.nih.gov/32594489/ Ince et al., 2019] || || Yes || |} == Limitations == When interpreting results from the BSDS, it is important to note that the BSDS has several limitations. The BSDS is an example of a self-report scale which relies on the individual’s subjective interpretation of their own symptoms and behaviours. An individual may consciously or subconsciously misrepresent the data due to a range of factors from [[w:Social-desirability bias|social desirability bias]] to faulty recall, which can compromise the accuracy of their BSDS score. An additional limitation is that the scale cannot confirm if an individual has bipolar disorder as it does not include all the signs of bipolar spectrum disorder listed by the [[w:DSM-5|DSM-5]]. A further limitation research studies are often conducted on small samples of outpatients, leading to varying scores of the accuracy and reliability of the BSDS.<ref name=":1" /> All these limitations may play some role in why the BSDS has been found to have such a low [[w:Positive Predictive Value|PPV]], leading to the overestimation of BD in individuals completing the scale. As such, it is important that the BSDS be used in conjunction with other clinical information to make a fully accurate diagnosis, but when used alone, the BSDS can have dangerous ramifications in overdiagnosing a serious psychiatric condition such as bipolar disorder to the general population. ==See also== * [[w:Bipolar disorder|Bipolar disorder]] * [[w:Rating scales for depression|Rating scales for depression]] * [[w:Mood Disorder Questionnaire|Mood Disorder Questionnaire]] * [[w:Diagnostic and Statistical Manual of Mental Disorders|Diagnostic and Statistical Manual of Mental Disorders]] * [[w:ICD-11|ICD-11]] ==References== {{reflist}} {{:{{BASEPAGENAME}}/Navbox}} [[w:Category:Bipolar disorder|Category:Bipolar disorder]] | [[w:Category:Depression (mood)|Depression (mood)]] | [[w:Category:Mental disorders screening and assessment tools|Mental disorders]] | [[w:Category:Mood disorders|Mood disorders]] | [[w:Category:Mania screening and assessment tools|Mania screening and assessment tools]] | [[w:Category:Treatment of bipolar disorder|Treatment of bipolar disorder]] h5bk3p3bxffjmj2h0om5ziipo12a2jv 0 304617 2691693 2677626 2024-12-12T22:18:31Z Dc.samizdat 2856930 Categorys 2691693 wikitext text/x-wiki {{Polyscheme}} Polyscheme is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in the higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[geometry]] in more than three dimensions."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project. [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop-up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|[[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the explanatory footnote, you can read a larger note like this one where it appears in the [[#Notes|Notes section]], below, like a mini-article within this article, which may occur in other Polyscheme project articles as well in some cases. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a maze of little tunnels! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in these companion articles is opinion, not established fact, as of this date of publication, and some of it is just commentary, not essential fact. The commentary and recent research is precisely the difference between the learning project article and the corresponding encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article, and are only a '''commentary''' on it. Participants are directed by a banner to "See also" the Wikipedia article when reading these commentaries. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery by a Swiss researcher around 1850. But for the first 50 years of its history [[W:Ludwig Schläfli|Ludwig Schläfli]]'s paper on the subject was unpublished, entirely inaccessible to other researchers. Even after its publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicans who could read that language. H.S.M Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been published since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the encyclopedia of [[W:Euclidean geometry|Euclidean geometry]], and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new chapters to it. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] 1snbli9kl863xs94sg7ysk1dc1cm1hb 2691703 2691693 2024-12-12T22:36:55Z Dc.samizdat 2856930 2691703 wikitext text/x-wiki {{Polyscheme}} Polyscheme is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in the higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[geometry]] in more than three dimensions."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project. [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop-up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|[[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the explanatory footnote, you can read a larger note like this one where it appears in the [[#Notes|Notes section]], below, like a mini-article within this article, which may occur in other Polyscheme project articles as well in some cases. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost [[W:https://en.wikipedia.org/wiki/Adventure_game|underground in a maze]] of little tunnels! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in these companion articles is opinion, not established fact, as of this date of publication, and some of it is just commentary, not essential fact. The commentary and recent research is precisely the difference between the learning project article and the corresponding encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article, and are only a '''commentary''' on it. Participants are directed by a banner to "See also" the Wikipedia article when reading these commentaries. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery by a Swiss researcher around 1850. But for the first 50 years of its history [[W:Ludwig Schläfli|Ludwig Schläfli]]'s paper on the subject was unpublished, entirely inaccessible to other researchers. Even after its publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicans who could read that language. H.S.M Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been published since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the encyclopedia of [[W:Euclidean geometry|Euclidean geometry]], and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new chapters to it. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] bm30oaxb33ml8jcnvc9byifjpkzpbnq 2691704 2691703 2024-12-12T22:37:50Z Dc.samizdat 2856930 2691704 wikitext text/x-wiki {{Polyscheme}} Polyscheme is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in the higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[geometry]] in more than three dimensions."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project. [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop-up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|[[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the explanatory footnote, you can read a larger note like this one where it appears in the [[#Notes|Notes section]], below, like a mini-article within this article, which may occur in other Polyscheme project articles as well in some cases. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost [[W:Adventure_game|underground in a maze]] of little tunnels! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in these companion articles is opinion, not established fact, as of this date of publication, and some of it is just commentary, not essential fact. The commentary and recent research is precisely the difference between the learning project article and the corresponding encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article, and are only a '''commentary''' on it. Participants are directed by a banner to "See also" the Wikipedia article when reading these commentaries. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery by a Swiss researcher around 1850. But for the first 50 years of its history [[W:Ludwig Schläfli|Ludwig Schläfli]]'s paper on the subject was unpublished, entirely inaccessible to other researchers. Even after its publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicans who could read that language. H.S.M Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been published since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the encyclopedia of [[W:Euclidean geometry|Euclidean geometry]], and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new chapters to it. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] q0dej9cqm3hum8jb2or286a4ngw88f2 2691705 2691704 2024-12-12T22:42:34Z Dc.samizdat 2856930 2691705 wikitext text/x-wiki {{Polyscheme}} Polyscheme is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in the higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[geometry]] in more than three dimensions."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project. [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be a series of wiki-format articles on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop-up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|[[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the explanatory footnote, you can read a larger note like this one where it appears in the [[#Notes|Notes section]], below, like a mini-article within this article, which may occur in other Polyscheme project articles as well in some cases. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a [[W:Twisty little maze of passages|twisty little maze of passages]]! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in these companion articles is opinion, not established fact, as of this date of publication, and some of it is just commentary, not essential fact. The commentary and recent research is precisely the difference between the learning project article and the corresponding encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article, and are only a '''commentary''' on it. Participants are directed by a banner to "See also" the Wikipedia article when reading these commentaries. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery by a Swiss researcher around 1850. But for the first 50 years of its history [[W:Ludwig Schläfli|Ludwig Schläfli]]'s paper on the subject was unpublished, entirely inaccessible to other researchers. Even after its publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicans who could read that language. H.S.M Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been published since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the encyclopedia of [[W:Euclidean geometry|Euclidean geometry]], and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new chapters to it. == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] tvhuhrrsr4x2qo4vehmor0e0i9xpu56 0 304718 2691694 2679932 2024-12-12T22:18:43Z Dc.samizdat 2856930 Categories 2691694 wikitext text/x-wiki {{Short description|Four-dimensional analog of the icosahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=600-cell| Image_File=Schlegel_wireframe_600-cell_vertex-centered.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]], vertex-centered<br>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Last=[[W:Rectified 600-cell|34]]| Index=35| Next=[[W:Truncated 120-cell|36]]| Schläfli={3,3,5}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}| Cell_List=600 ([[W:Tetrahedron|{3,3}]]) [[Image:Tetrahedron.png|20px]]| Face_List=1200 [[W:triangle|{3}]]| Edge_Count=720| Vertex_Count= 120| Petrie_Polygon=[[W:Triacontagon#Petrie polygons|30-gon]]| Coxeter_Group=H₄, [3,3,5], order 14400| Vertex_Figure=[[Image:600-cell verf.svg|80px]]<br>[[W:icosahedron|icosahedron]]| Dual=[[120-cell|120-cell]]| Property_List=[[W:Convex polytope|convex]], [[W:isogonal figure|isogonal]], [[W:isotoxal figure|isotoxal]], [[W:isohedral figure|isohedral]] }} [[File:600-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[geometry]], the '''600-cell''' is the [[W:convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,5}. It is also known as the '''C₆₀₀''', '''hexacosichoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hexacosihedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and a '''[[W:polytetrahedron|polytetrahedron]]''', being bounded by tetrahedral [[W:Cell (geometry)|cells]]. The 600-cell's boundary is composed of 600 [[W:Tetrahedron|tetrahedral]] [[W:Cell (mathematics)|cells]] with 20 meeting at each vertex.{{Efn|name=vertex icosahedral pyramid}} Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:icosahedron|icosahedron]], since it has five [[W:Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[W:triangle|triangle]]s meeting at every vertex.{{Efn|name=math of dimensional analogy}} Its [[W:dual polytope|dual polytope]] is the [[120-cell|120-cell]]. == Geometry == The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|name=4-polytopes ordered by size and complexity|group=}} It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell|24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[W:Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), "600-cell" column ₀''R/l'' {{=}} 2𝝓/2}} which is the [[W:golden ratio|golden ratio]]. {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === ==== Unit radius Cartesian coordinates ==== The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length {{sfrac|1|φ}} ≈ 0.618 (where φ = {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows: 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates, and 16 vertices of the form: :(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) The remaining 96 vertices are obtained by taking [[W:even permutation|even permutation]]s of :(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ⁻¹|2}}, 0) Note that the first 8 are the vertices of a [[16-cell|16-cell]], the second 16 are the vertices of a [[W:tesseract|tesseract]], and those 24 vertices together are the vertices of a [[24-cell|24-cell]]. The remaining 96 vertices are the vertices of a [[W:snub 24-cell|snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} When interpreted as [[W:#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[W:icosian|icosian]]s. In the 24-cell, there are [[24-cell#Great squares|squares]], [[24-cell#Great hexagons|hexagons]] and [[24-cell#Triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}} In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}} In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]]. The 60 axes and 75 16-cells of the 600-cell constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60₅75₄ to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}} Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs. ==== Hopf spherical coordinates ==== In the 600-cell there are also great circle [[W:pentagon|pentagon]]s and [[W:decagon|decagon]]s (in central planes through ten vertices).{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[W:#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}} Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[W:#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}} By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉_''i'', 𝜂, 𝜉_''j''){{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles. A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two [[W:completely orthogonal|completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<br> : w {{=}} cos 𝜉_''i'' sin 𝜂 : x {{=}} cos 𝜉_''j'' cos 𝜂 : y {{=}} sin 𝜉_''j'' cos 𝜂 : z {{=}} sin 𝜉_''i'' sin 𝜂 The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as: : ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}) where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉_''i'' and 𝜉_''j'' coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}} === Structure === ==== Polyhedral sections ==== The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[W:hypersphere|hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[W:icosahedron|icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[W:dodecahedron|dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[W:icosidodecahedron|icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex}}{{Sfn|Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout|Parker|1998}} These can be seen in the H3 [[W:Coxeter plane|Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} :[[File:600-cell-polyhedral levels.png|640px]] These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron. {| class=wikitable !colspan=2|Concentric Hulls |- |align=center|[[Image:Hulls of H4only-orthonormal.png|360px]] |The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br> <br> 1) two points at the origin<br> 2) two icosahedra<br> 3) two dodecahedra<br> 4) two larger icosahedra<br> 5) and a single icosidodecahedron<br> <br> for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices. |- |} ==== Golden chords ==== [[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]] The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]]. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]] {{see also|W:24-cell#Hypercubic chords|label 1=24-cell § Hypercubic chords}} The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex; see column ''a''}} at eight different [[W:Chord (geometry)|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}} In ascending order of length, they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=The fractional square roots are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <math>\tfrac{1}{\phi} = \phi^{-1}</math> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² ≈ 0.382<br> For example: {{indent|7}}𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618|name=fractional square roots|group=}} Notice that the four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] <big>φ</big> {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to <big>𝜋</big>:<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=W:John Carlos Baez|access-date=10 October 2022}}</ref><br> : {{sfrac|𝜋|5}} = arccos ({{sfrac|φ|2}}) is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord. Reciprocally, in this function discovered by Robert Everest expressing <big>φ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br> : <big>φ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}}) {{sfrac|3𝜋|5}} is the arc length of the <big>φ</big> = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge. The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is <big>φ</big>, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the <big>φ</big> chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}} The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}. The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <big>φ</big>{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}} ==== Boundary envelopes ==== [[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]. The 3D surface made of 600 tetrahedra is visible.]] The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a [[W:snub 24-cell|snub 24-cell]]. Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}} The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: tetrahedra of edge length {{sfrac|1|φ}} ≈ 0.618 instead of octahedra of edge length 1. It encloses the {{radic|1}} edges of the 24-cells, which become invisible interior chords in the 600-cell, like the [[24-cell#Hypercubic chords|{{radic|2}} and {{radic|3}} chords]]. [[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell|24-cell]] performing a [[24-cell#Simple rotations|simple rotation]]. The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.]] Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of {{sfrac|1|φ}}, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[W:Tesseract#Radial equilateral symmetry|radially equilateral]]. Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[W:equilateral triangle|equilateral triangle]]s which all meet at the center.{{Efn|The long radius (center to vertex) of the [[24-cell#geometry|24-cell]] is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[W:Golden triangle (mathematics)|golden triangle]]s which all meet at the center.{{Efn|A [[W:Golden triangle (mathematics)|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ratio|golden ratio]] to the distinct side ''b'': : {{sfrac|a|b}} <nowiki>=</nowiki> φ <nowiki>=</nowiki> {{sfrac|1 + {{radic|5}}|2}} <nowiki>≈</nowiki> 1.618 It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center, and in the regular [[W:Pentagon|pentagon]] by connecting any two adjacent vertices to the vertex opposite them.<br> The vertex angle is: : <nowiki>𝛉 = arccos(</nowiki>{{sfrac|φ|2}}<nowiki>) = </nowiki>{{sfrac|𝜋|5}}<nowiki> = 36°</nowiki> so the base angles are each {{Sfrac|2𝜋|5}} <nowiki>=</nowiki> 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=Golden triangle}} All the [[W:regular polytope|regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[W:Schläfli orthoscheme|orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[W:right triangle|right triangle]]s which meet at the center.{{Efn|The [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is the generalization of the [[W:right triangle|right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[W:golden ratio|golden ratio]] to its edge length; thus its radius is <big>φ</big> if its edge length is 1, and its edge length is {{sfrac|1|φ}} if its radius is 1.}} Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[W:icosidodecahedron|icosidodecahedron]], and the two-dimensional [[W:Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) '''Radially golden''' polytopes are those which can be constructed, with their radii, from [[W:Golden triangle (mathematics)|golden triangles]].{{Efn|name=Golden triangle}} The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an [[W:Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[W:#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} Therefore the successor may be constructed by placing [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[W:cubic pyramid|cubic pyramid]]s. But if we place 24 canonical [[W:octahedral pyramid|octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}} ==== Geodesics ==== The vertex chords of the 600-cell are arranged in [[W:geodesic|geodesic]] [[W:great circle|great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§4 The planes of the 600-cell|pp=437-439}} [[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[W:stereographic projection|stereographic projection]] of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]] The {{radic|0.𝚫}} = 𝚽 edges form 72 flat regular central [[W:decagon|decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}} Just as the [[W:icosidodecahedron|icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 {{radic|0.𝚫}} edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, {{radic|3.𝚽}} apart. As in the decagon and the icosidodecahedron, the edges occur in [[W:Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope. The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[W:Hopf fibration|Hopf fibration]]s, each filling the whole 600-cell. Each [[W:#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} The {{radic|1}} chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}} Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The {{radic|1}} chords join vertices which are two {{radic|0.𝚫}} edges apart. Each {{radic|1}} chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[W:triangular bipyramid|triangular bipyramid]]), and passes through the center of the shared face. As there are 1200 faces, there are 1200 {{radic|1}} chords, in 600 parallel pairs, {{radic|3}} apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell|Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}} The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} The {{radic|1.𝚫}} chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}} The {{radic|1.𝚫}} chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. The {{radic|1.𝚫}} chords join vertices which are two {{radic|0.𝚫}} edges apart on a geodesic great circle. The 720 {{radic|1.𝚫}} chords occur in 360 parallel pairs, {{radic|2.𝚽}} = φ apart. The {{radic|2}} chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex). The {{radic|2}} chords join vertices which are three {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart). There are 600 {{radic|2}} chords, in 300 parallel pairs, {{radic|2}} apart. The 450 great squares (225 [[W:Completely orthogonal|completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} The {{radic|2.𝚽}} = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length {{radic|3.𝚽}}. The {{radic|2.𝚽}} chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 720 distinct {{radic|2.𝚽}} chords, in 360 parallel pairs, {{radic|1.𝚫}} apart. The {{radic|3}} chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 {{radic|3}} chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The {{radic|3}} chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The {{radic|3}} chords join vertices which are four {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart on a geodesic great circle). Each {{radic|3}} chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 {{radic|1}} cubic cells. The 1200 {{radic|3}} chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each {{radic|3}} chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}} There are 1200 {{radic|3}} chords, in 600 parallel pairs, {{radic|1}} apart. The {{radic|3.𝚽}} chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length {{radic|1.𝚫}}, so these are [[W:Golden triangle (mathematics)|golden triangles]]. The {{radic|3.𝚽}} chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 720 distinct {{radic|3.𝚽}} chords, in 360 parallel pairs, {{radic|0.𝚫}} apart. The {{radic|4}} chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell. The {{radic|4}} chords join opposite vertices which are five {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells. The sum of the squared lengths{{Efn|The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.}} of all these distinct chords of the 600-cell is 14,400 = 120².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[W:completely orthogonal|completely orthogonal]] to a great 30-gon{{Efn|A ''[[W:triacontagon|triacontagon]]'' or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the [[W:Interior angle|interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[W:Equilateral triangle|equilateral triangle]] (60°) and the [[W:Regular pentagon|regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. It is the central axis of a spiral skew 30-gram, the [[W:Petrie polygon|Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}} Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}} All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:icosidodecahedron|icosidodecahedron]]. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens|Seidel|1973}} Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together. Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration. Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles. Vectors in 4-space may be combined by [[W:Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[W:William Rowan Hamilton|Hamilton]].{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}} Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}} Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}} Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}} Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}} Each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one {{radic|4}} long diameter): a great [[W:digon|digon]] plane.{{Efn|In the 24-cell each great square plane is [[W:completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:digon|digon]] plane.|name=digon planes}} Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere. Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell. In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}} ==== Fibrations of great circle polygons ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[W:fiber bundle|fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[W:Hopf fibration|Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}} The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell|Semple|1971|loc=§4. Isoclinic planes in Euclidean space E₄|pp=6-7}} ===== Decagons ===== [[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[W:Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[W:Schläfli double six|Schläfli double six]] configuration 30₂12₅ characteristic of the H₄ polytopes. The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]] The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons. Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[W:icosahedron|icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}} Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}} The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[W:Hopf fibration|Hopf fibration]]s make the 600-cell a [[W:Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30₂12₅. It is called the [[W:Schläfli double six|Schläfli double six]] configuration after [[W:Ludwig Schläfli|Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[W:Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[W:Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3^{2, 2, 1}] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} ===== Hexagons ===== The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration. The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. The Hopf map of this fibration is the [[W:dodecahedron|dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations. The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells. ===== Squares ===== The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the {{radic|2}} tetrahedral cells of the 75 inscribed 16-cells, ''not'' the {{radic|0.𝚫}} tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} The Hopf map of this fibration is the [[W:icosidodecahedron|icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations. ===== Clifford parallel cell rings ===== The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, studied cell rings in the general case of their geometry and [[W:group theory|group theory]], identifying each cell ring as a [[W:polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]].{{Efn|name=orthoscheme ring}} He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:Chiral|chiral]] forms. Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}} Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}} The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}} This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[W:Hopf fibration|Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}} The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}} The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with ''subfibrations''. The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings. The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}} The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope.}} The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[W:Hopf fibration|Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[W:2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}} On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is the [[W:icosahedron|icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration. But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[W:Vertex figure|vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[W:Braid|braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Sfn|Sadoc|Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,{{Efn|name=Clifford parallels}} along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint."}}|name=Hopf fibration base}} of these fibrations on the 2-sphere). Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 {{Background color|red}} faces of the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration''). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}} === Constructions === The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize. ==== Gosset's construction ==== [[W:Thorold Gosset|Thorold Gosset]] discovered the [[W:Semiregular polytope|semiregular 4-polytopes]], including the [[W:Snub 24-cell|snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described [[W:Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[W:Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}} The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}} The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell. Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps. The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[W:Midsphere|midsphere]] to construct its outer [[W:Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell. ==== Cell clusters ==== Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]], or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.{{Efn|name=tetrahedral cell adjacency}} Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional space that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[W:Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point. ===== Icosahedra ===== [[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[W:Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[W:Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[W:Regular icosahedron#Symmetries|snub octahedron]]. This term refers specifically to a [[W:Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}} Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the [[W:Icosahedral pyramid|icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.|alt=|200x200px]] The [[W:Vertex figure|vertex figure]] of the 600-cell is the [[W:Icosahedron|icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical [[W:Icosahedral pyramid|icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|name=radially equilateral icosahedral pyramid}}|name=vertex icosahedral pyramid|group=}} Twenty tetrahedral cells meet at each vertex, forming an [[W:Icosahedral pyramid|icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron. It is remarkable that twenty regular tetrahedra fit inside a regular icosahedral pyramid in 4-space. In 3-space, twenty tetrahedral pyramids fit inside a regular icosahedron around its center but they are ''not'' regular tetrahedra, because the regular icosahedron's radius is not the same as its edge length.{{Efn|In Euclidean 3-space, the icosahedron is not [[W:Cuboctahedron#Radial equilateral symmetry|radially equilateral like the cuboctahedron]]. The icosahedron's radii are shorter than its edge length. But in the [[W:3-sphere|spherical 3-space]] of the 600-cell's surface the center of a regular icosahedron is lifted orthogonally out of its 3-space hyperplane: remarkably, just far enough to make its radii the same length as its edges. As a figure in Euclidean 4-space, this radially equilateral spherical icosahedron is an [[W:Icosahedral pyramid|icosahedral pyramid]]. In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not its center, just one of its vertices. But in curved 3-space the 12 edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that spherical space''. In Euclidean 4-space there are only two radially equilateral figures: 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[W:tesseract#Radial equilateral symmetry|radially equilateral tesseract]].|name=radially equilateral icosahedral pyramid}} The 600-cell has a [[W:Dihedral angle|dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}} An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}} The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[W:Snub 24-cell|snub 24-cell]], which has exactly the same [[W:Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing. The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[W:Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}} Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a [[24-cell#Great hexagons|great circle hexagon]]. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[W:Triangular bipyramid|triangular bipyramid]]) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a {{radic|1}} octahedral cell, but in the larger {{radic|2}} octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]]. The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}} The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[W:Russian dolls|Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}} The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}} The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[W:Jessen's icosahedron|Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron;{{Sfn|Itoh|Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[W:Jitterbug transformation|Jitterbug]] by [[W:Buckminster Fuller|Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}} The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[W:Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca|Al-Mukhaini|Koca|Al Qanobi|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[W:3-sphere|S³]] is similar to the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]] and the [[W:Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[W:Three-dimensional space|R³]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[W:Buckminster Fuller|Buckminster Fuller]].<ref>{{cite journal | last = Verheyen | first = H. F. | doi = 10.1016/0898-1221(89)90160-0 | issue = 1–3 | journal = [[W:Computers and Mathematics with Applications|Computers and Mathematics with Applications]] | mr = 0994201 | pages = 203–250 | title = The complete set of Jitterbug transformers and the analysis of their motion | volume = 17 | year = 1989| doi-access = free }}</ref>}} The tetrahedral cells are face-bonded into [[W:Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[W:Boerdijk-Coxeter helix|Boerdijk-Coxeter helix]]. This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]]. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}} 5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[W:Hopf fibration|Hopf fibration]]. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}} The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other. Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[W:Skew lines|skew edges]] of the tetrahedron. Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[W:Hopf fibration|Hopf fiber bundle]] of 12. Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]]. The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}} Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} ===== Octahedra ===== There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell. Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell. First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[W:Triangular dipyramid|triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length {{radic|1}}. Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length {{radic|1}}. They form a tetrahedron of edge length {{radic|1}}, which is the second section of the 600-cell beginning with a cell.{{Efn|The {{radic|1}} tetrahedron has a volume of 9 {{radic|0.𝚫}} tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the {{radic|1}} tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}} There are 600 of these {{radic|1}} tetrahedral sections in the 600-cell. With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length {{radic|1}}, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}} As partially filled so far (by 17 tetrahedral cells), this {{radic|1}} octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each {{radic|1}} edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}} Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A {{radic|1}} octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same {{radic|1}} octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point {{radic|1}} octahedral section, a 4-point {{radic|1}} tetrahedral section, and a 4-point {{radic|0.𝚫}} tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the {{radic|1}} octahedron surrounds the {{radic|1}} tetrahedron which surrounds the {{radic|0.𝚫}} tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[W:Octahedral pyramid|octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}} Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical {{radic|1}} [[W:Octahedral pyramid|octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter {{radic|0.𝚫}} edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a {{radic|1}} tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with {{radic|0.𝚫}} edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two {{radic|1}} edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three {{radic|0.𝚫}} edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[W:Rectified 5-cell|rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length {{sfrac|1|φ}} ≈ 0.618. ===== Union of two tori ===== There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell. An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines". [[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[W:Clifford torus|Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}} Its opposite edges are identified, forming a [[W:Duocylinder|duocylinder]].]] The [[120-cell|120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]]. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}} Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex-stacked [[W:Icosahedral pyramids|icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This 10-cell ring is shaped like a [[W:Pentagonal antiprism|pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center. This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}} Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}} The surface is the same as that of ten stacked [[W:pentagonal antiprism|pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[W:Triangular bipyramid|triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge. The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}} The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes). Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the [[W:Duocylinder|duocylinder]] and form a [[W:Clifford torus|Clifford torus]].{{Efn|A [[W:Clifford torus|Clifford torus]] is the [[W:Hopf fibration|Hopf fiber bundle]] of a distinct [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[W:3-sphere|3-sphere]], involving all of its points. The [[W:SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[W:Cartesian product|Cartesian product]] of two [[W:Completely orthogonal|completely orthogonal]] [[W:Great circle|great circle]]s. It is a filled [[W:Doughnut|doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[W:4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}} They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the [[W:Tetrahedral-octahedral honeycombtetrahedral-octahedral honeycomb]]. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} But how can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[W:Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}} In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[W:Triangular bipyramid|triangular bipyramid]] composed of two tetrahedra. This decomposition of the 600-cell has [[W:Coxeter notation|symmetry]] [[W:10,2⁺,10|10,2⁺,10]], order 400, the same symmetry as the [[W:Grand antiprism|grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[W:Pentagonal antiprism|pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}} The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells. ===== Boerdijk–Coxeter helix rings ===== The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[W:Hopf fibration|Hopf fibration]] which fills the entire 600-cell.{{Sfn|Banchoff|1988}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}} Each ring of 30 face-bonded tetrahedra is a cylindrical [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension. {| class="wikitable" width="600" |[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}} |[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}} |[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[W:Skew polygon|skew]] {30/11} [[W:Star polygon|star polygon]] with a [[W:Winding number|winding number]] of 11 called a [[W:Triacontagon#Triacontagram|triacontagram₁₁]], a continuous tight corkscrew [[W:Helix|helix]] bent into a loop of 30 edges (the {{Background color|magenta|magenta}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[W:Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[W:Density (polytope)|characterized]] as the [[W:Petrie polygon|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}} |- |colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''. |} The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three {{Background color|cyan|cyan}} Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}} The 30 {{Background color|magenta|magenta}} edges joining these vertex pairs form a helical [[W:Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[W:Petrie polygon|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[W:Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[W:Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h₁ h₂''.}}|name=Petrie polygon in 30-cell ring}} The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[W:Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell|120-cell]], the 600-cell's [[W:Dual polytope|dual polytope]].{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the 600-cell and its dual the [[120-cell|120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}} The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices. Each orange or yellow edge crosses between two {{Background color|cyan|cyan}} great decagons. Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}} Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[W:Geodesic|geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}} The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between. A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop. The two connected loops forms a single [[W:Möbius loop|Möbius loop]], a skew {15/2} [[W:Pentadecagram|pentadecagram]]. The pentadecagram is not shown in these illustrations (but [[#Decagons and pentadecagrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations. Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}} Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}} They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[W:link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}} Five of these 30-cell [[W:Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[W:Chiral|chiral]] objects; they either spiral clockwise (right) or counterclockwise (left). The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings. Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[W:Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]]. Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration. In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings. Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[W:Hopf link|Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Efn|Sadoc describes the decomposition of the 600-cell into four tori.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} It is the same [[#Decagons|fibration of 12 great decagons and 20 30-cell rings]], seen as a [[#Clifford parallel cell rings|fibration of four completely disjoint 30-cell rings]]{{Efn|name=completely disjoint}} with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we ''can'' discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis, and 6 decagons each. Their axis is a 30-cell ring, not a decagon, and they contain only 3 decagons each. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated ("bumpy") rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring "holes" left to fill in the 600-cell. To do that, fill some of the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both. On the icosahedron Hopf map,{{Efn|name=Hopf fibration base}} the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of "extra flap" like the sixth square flap of the [[W:Cube#Orthogonal projections|net of a cube]] before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus ''every'' 30-cell ring is the center core of a 150-cell ring.}} ==== Radial golden triangles ==== The 600-cell can be constructed radially from 720 [[W:Golden triangle (mathematics)|golden triangle]]s of edge lengths {{radic|0.𝚫}} {{radic|1}} {{radic|1}} which meet at the center of the 4-polytope, each contributing two {{radic|1}} radii and a {{radic|0.𝚫}} edge. They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral {{radic|0.𝚫}} bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular {{radic|0.𝚫}} tetrahedron bases (the cells of the 600-cell). ==== Characteristic orthoscheme ==== {| class="wikitable floatright" !colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\tfrac{1}{\phi} \approx 0.618</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |align=center|<small>164°29′</small> |align=center|<small><math>\pi-2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}} \approx 0.951</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length <small><math>\text{𝒍} = \tfrac{1}{\phi} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center. ==== Reflections ==== The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}} For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram₂ geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}} Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Weyl orbits ==== Another construction method uses [[#Symmetries|quaternions]] and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4): : O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} : O(1000) : V1 : O(0010) : V2 : O(0001) : V3 [[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]] With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell|120-cell]] of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the [[120-cell|120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> === Rotations === The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one [[W:Completely orthogonal|completely orthogonal]] invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]].{{Efn|name=isoclinic geodesic}} The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge-lengths away.{{Efn|name=isoclinic rotation to non-adjacent vertices}} For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex {{radic|1}} (60°) distant, moving {{radic|1/4}} {{=}} 1/2 unit radius in four orthogonal directions.|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles. These '''[[#Rotations on polygram isoclines|isoclines]]''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere]|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in [[W:Chiral|chiral]] pairs as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true 5𝝅 circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}} The 600-cell is generated by [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes{{Efn|name=isoclinic invariant planes}} are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:William Kingdon Clifford|Clifford]] displacement is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn||name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane rotates sideways.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} ''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}} ==== Twenty-five 24-cells ==== There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney|Hooker|Johnson|Robinson|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.{{Efn|name=Six orthogonal planes of the Cartesian basis}} The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a {{radic|3}} chord.|name=120° apart}} The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell|Aravind|2009|loc=§3. The 600-cell|pp=2-5}} There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[W:Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}}} Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]]. The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane. There are an infinite number of pairs of [[W:Completely orthogonalcompletely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}} Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel [[W:Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long. The three Clifford parallel decagons can be seen as the {{Background color|cyan}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]. Each {{Background color|magenta}} edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}} An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}} Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}} The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, and four are left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}|name=rotations to 8 disjoint 24-cells}} The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}} Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}} Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}} With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}} Non-disjoint 24-cells are related by a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane. They are also related by an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[W:Digon|digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}} There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}} As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space. Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}} ''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]]. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}} ''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart. However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}} Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}} (There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}} Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}} In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}. The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}} On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]]. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel. ==== Rotations on polygram isoclines ==== The regular convex 4-polytopes each have their characteristic kind of right (and left) [[W:Isoclinic rotation|isoclinic rotation]], corresponding to their characteristic kind of discrete [[W:Hopf fibration|Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[W:Completely orthogonal|completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.''{{Efn|Consider the statement: ''In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber.'' It can be found in the literature, expressed in the mathematical language of the Hopf fibration,{{Sfn|Kim|Rote|2016|loc= 8 The Construction of Hopf Fibrations|pp=12-16|ps=; see Theorem 13.}} but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting sideways, rotating in parallel with the completely orthogonal great circle.{{Efn|name=isoclinic geodesic}} With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.{{Efn|name=helical geodesic}} Each helical isocline is itself a kind of circle, but it is not a planar great circle of the [[W:Hopf fibration|Hopf fibration]]: it is a special kind of geodesic circle whose circumference is greater than 2𝝅''r'', and it is not pictured explicitly at all by the plain statement we are trying to visualize. We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does ''not'' simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle ''as every great circle itself is moving orthogonally'', flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous ''twisting'' rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once ''in a full isoclinic revolution, every vertex moves more than 360 degrees,'' as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the [[24-cell#Isoclinic rotations|isoclinic rotation of a great hexagon fibration of the 24-cell]], each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle);{{Efn|name=4𝝅 rotation}} but in the [[#Decagons and pentadecagrams|isoclinic rotation of a great decagon fibration of the 600-cell]], each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).}} The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[W:Chain mail|chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell. Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[W:Skew polygon|skew]] ''[[W:Polygram (geometry)|polygrams]]'' (which trace the paths on the [[W:Clifford torus|Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[W:Fiber bundle|fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}} Since isoclinic rotations are [[W:Chiral|chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}} All the fiber bundles are aspects of the same discrete [[W:Hopf fibration|Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations. Cell rings are another expression of the Hopf fibration. Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}} The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells. The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets. Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[W:Loom|loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers. A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[W:Warp and woof|warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}} In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia|Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}} The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[W:Chiral|chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[W:Chiral|chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}} The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[W:Torsion of a curve|torsion]]). Pairs of left-handed and right-handed isoclines run through cell rings of this kind. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}} Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively. Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once. It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers. Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating in "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} Left and right isoclines are different paths that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers. A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation. Each moving vertex travels along an isocline contained within a (moving) cell ring. While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}} The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations. Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines. The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}} A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[W:#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.) In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration. An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices (two or more edge-lengths away){{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex at least two edge-lengths away. In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is exactly two edge-lengths away along one of several great circle geodesic routes: the opposite vertex of a neighboring cell. In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron. |name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind). A left-right pair of isoclinic rotations constitutes a discrete fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}} The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[W:Geodesic|geodesic]]. ==== Decagons and pentadecagrams ==== The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines. The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}} 12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[W:Hopf fibration|Hopf fibration]]. There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}} The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons. The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]]. A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration). The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether. Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices. In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}} In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon. In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a [[W:Hexagram|hexagram]]: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are {{radic|3}} chords of the hexagon instead of {{radic|1}} hexagon edges.{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=4-polytopes ordered by size and complexity}} In the {{radic|1}} [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply {{radic|3}} chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic {6/2} hexagram rotation]] both rotate circles of 6 vertices. The hexagram isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}} The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of hexagrams]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}} In the {{radic|2}} [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its {{radic|2}} edges and its {{radic|4}} diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[W:Winding number|winds]] ''three'' times around the 3-sphere. The invariant central plane completely orthogonal to each {4} great square is another {4} great square {{radic|4}} distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares. The 16-cell's [[W:Dural polytope|dual polytope]] the [[W:8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). In the 8-cell this is a rotation of {{radic|1}} × {{radic|3}} great rectangles, and also a rotation of {{radic|4}} axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]]. In the {{radic|0.𝚫}} [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are {{radic|1}} hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}} The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices. The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices. The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great [[W:Digon|digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell. The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis. The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=4-polytopes ordered by size and complexity}} nested like [[W:Russian dolls|Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}} In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it. Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}} The two non-adjacent vertices are linked by a {{radic|1}} chord of the isocline which is a great hexagon edge (a 24-cell edge). The {{radic|1}} chords of the 30-cell ring (without the {{radic|0.𝚫}} 600-cell edges) form a skew [[W:Triacontagram|triacontagram]]_{{30/4}=2{15/2}} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[W:Pentadecagram|pentadecagram]]₂ isoclines. Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers. Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[W:Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]], '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. (Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.) Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}} Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}} The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations. At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}} Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines. There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell. Each distinct isocline belongs to 5 fiber bundles. {| class="wikitable" width="450" !colspan=4|Three sets of 30-cell ring chords from the same [[W:Orthogonal projection|orthogonal projection]] viewpoint |- ![[W:Pentadecagon#Pentadecagram|Pentadecagram {15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]] |- |colspan=2 align=center|All edges are [[W:Pentadecagram|pentadecagram]] isocline chords of length {{radic|1}}, which are also [[24-cell#Great hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell. |colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length {{radic|1.𝚫}} ≈ 1.176. |- |[[File:Regular_star_polygon_15-2.svg|200px]] |[[File:Regular_star_figure_2(15,2).svg|200px]] |[[File:Regular_star_figure_6(5,1).svg|200px]] |- |valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The {{radic|1}} chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart. |valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The {{radic|1}} chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra. |valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly congruent, each ''acting'' as a left or right isocline in different fibrations. |- |colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. |} Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings, but the two isoclines in each 3-cell ring are directly congruent.{{Efn|The chord-path of an isocline may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Boerdijk–Coxeter helix rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has two helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}} The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one {{radic|0.𝚫}} edge-length apart). The 30 chords joining the isocline's 30 vertices are {{radic|1}} hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell {{radic|0.𝚫}} edges apart on a decagon great circle. {{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram₂ geodesic]] is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions. The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon. The Petrie triacontagon has {{radic|0.𝚫}} edges; the isoclinic pentadecagram₂ has {{radic|1}} edges which join vertices which are two {{radic|0.𝚫}} edges apart. Each {{radic|1}} edge belongs to a different [[#Hexagons|great hexagon]], and successive {{radic|1}} edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}} These isocline chords are both hexa''gon'' edges and penta''gram'' edges. The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other. Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[W:Chessboard|chessboard]].{{Efn|name=isoclinic chessboard}} In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring. ==== Hexagons and hexagrams ==== [[File:Regular_star_figure_2(10,3).svg|thumb|[[W:Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines. Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black {{radic|3}} [[24-cell#Triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring. The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 {{radic|1.𝚫}} chords of each isocline form a skew [[W:Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[W:Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[W:Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]]. Notice the relation between the [[24-cell#Helical hexagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on hexagram isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline because the 600-cell contains multiple 24-cells, and the 600-cell's {{radic|1.𝚫}} isocline chord is shorter than the 24-cell's {{radic|3}} isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on [[24-cell#Helical hexagrams and their 4𝝅 isoclines|hexagrams]], while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference 2𝝅; simple rotations take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[W:20-gon|icosagram]] is a compound of the 24-cell's helical {6/2} hexagram, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}} ==== Squares and octagrams ==== [[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[W:24-gon#Related polygons|{24/5} 24-gram]], with <big>φ</big> {{=}} {{radic|2.𝚽}} edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell ({{radic|1}} edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines. Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines. The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <big>φ</big> {{=}} {{radic|2.𝚽}} chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one {{radic|1}} chord apart, and 5 <big>φ</big> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell. Notice the relations between the [[16-cell#Helical construction|16-cell's rotation of just 2 invariant great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the {{radic|4}} diameter. In the 600-cell vertices are closer together, and its {{radic|2.𝚽}} {{=}} <big>φ</big> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} ! [[W:k-face|''k''-face]]||f_''k''||f₀ || f₁||f₂||f₃||[[W:Vertex figure|''k''-fig]] !Notes |- align=right |H₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|5|node}} ||( ) !f₀ || 120 || 12 || 30 || 20 ||[[W:icosahedron|{3,5}]] || H₄/H₃ = 14400/120 = 120 |- align=right |A₁H₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|5|node}} ||{ } !f₁ || 2 || 720 || 5 || 5 || [[W:pentagon|{5}]] || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |A₂A₁ ||{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node}} ||[[W:equilateral triangle|{3}]] !f₂ || 3 || 3 || 1200 || 2 || { } || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |A₃ ||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}} ||[[W:tetrahedron|{3,3}]] !f₃ || 4 || 6 || 4 || 600|| ( ) || H₄/A₃ = 14400/24 = 600 |} == Symmetries == The [[W:Icosian|icosian]]s are a specific set of Hamiltonian [[W:Quaternion|quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}} The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[W:Rational number|rational number]]s.{{Sfn|Steinbach|1997|p=24}} The finite sums of the 120 [[W:Icosian#Unit icosians|unit icosians]] are called the [[W:Icosian#Icosian ring|icosian ring]]. When interpreted as quaternions,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the 120 vertices of the 600-cell form a [[W:group (mathematics)|group]] under quaternionic multiplication. This group is often called the [[W:Binary icosahedral group|binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[W:Icosahedral group|icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}} It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[W:Invariant subgroup|invariant subgroup]], namely as the subgroup ''2I_L'' of quaternion left-multiplications and as the subgroup ''2I_R'' of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of ''2I_L'' and ''2I_R''; the pair of opposite elements generate the same element of ''RSG''. The [[W:Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''. We have the isomorphism ''RSG ≅ (2I_L × 2I_R) / {Id, -Id}''. The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200. The [[W:Quaternion algebra|quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[W:Rotations in 4-dimensional Euclidean spacerotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|2015|p=1|loc="''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}} The binary icosahedral group is [[W:Isomorphic|isomorphic]] to [[W:special linear group|SL(2,5)]]. The full [[W:Symmetry group|symmetry group]] of the 600-cell is the [[W:H4 (mathematics)|Coxeter group H₄]].{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§2 The Labeling of H₄}} This is a [[W:Group (mathematics)|group]] of order 14400. It consists of 7200 [[W:Rotation (mathematics)|rotations]] and 7200 rotation-reflections. The rotations form an [[W:Invariant subgroup|invariant subgroup]] of the full symmetry group. The rotational symmetry group was first described by S.L. van Oss.{{Sfn|Oss|1899||pp=1-18}} The H₄ group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} == Visualization == The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}} One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections. === 2D projections === The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" width=600 |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:600-cell graph H4.svg|200px]]<br>[30]<br>(Red=1) |[[File:600-cell t0 p20.svg|200px]]<br>[20]<br>(Red=1) |[[File:600-cell t0 F4.svg|200px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:600-cell t0 H3.svg|200px]]<br>[10]<br>(Red=1,orange=5,yellow=10) |[[File:600-cell t0 A2.svg|200px]]<br>[6]<br>(Red=1,orange=3,yellow=6) |[[File:600-cell t0.svg|200px]]<br>[4]<br>(Red=1,orange=2,yellow=4) |} === 3D projections === A three-dimensional model of the 600-cell, in the collection of the [[W:Institut Henri Poincaré|Institut Henri Poincaré]], was photographed in 1934–1935 by [[W:Man Ray|Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.<ref>{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&nbsp;58; ''Antony and Cleopatra'', SE-6, p.&nbsp;59; ''mathematical object mo-9'', p.&nbsp;64; ''Merchant of Venice'', SE-9, p.&nbsp;65, and "The Hexacosichoron", Philip Ordning, p.&nbsp;96.</ref> {| class=wikitable !colspan=2|Vertex-first projection |- |[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]] |This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied: * The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown. * The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image. |- !colspan=2|Cell-first projection |- |[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]] |This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied: * The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image. * The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint have been culled for clarity. This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. |} {| class=wikitable !Frame synchronized orthogonal isometric (left) and perspective (right) projections |- |[[File:Cell600Cmp.ogv|640px]] |} == Diminished 600-cells == The [[W:Snub 24-cell|snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell|24-cell]] and taking the [[W:Convex hull|convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[W:Diminishment (geometry)|diminishing]]'' of the 600-cell. The [[W:Grand antiprism|grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} A bi-24-diminished 600-cell, with all [[W:Tridiminished icosahedron|tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells. There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.<ref>{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}</ref> {| class="wikitable collapsible" !colspan=12|Diminished 600-cells |- !Name !Tri-24-diminished 600-cell !Bi-24-diminished 600-cell ![[W:Snub 24-cell|Snub 24-cell]]<br>(24-diminished 600-cell) ![[W:Grand antiprism|Grand antiprism]]<br>(20-diminished 600-cell) !600-cell |- align=center !Vertices |48 |72 |96 |100 |120 |- align=center !Vertex figure<br>(Symmetry) |[[File:Dual tridiminished icosahedron.png|120px]]<br>dual of tridiminished icosahedron<br>([3], order 6) |[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]<br>[[W:Hexahedron|tetragonal antiwedge]]<br>([2]⁺, order 2) |[[File:Snub 24-cell verf.png|120px]]<br>[[W:tridiminished icosahedron|tridiminished icosahedron]]<br>([3], order 6) |[[File:Grand antiprism verf.png|120px]]<br>[[W:Edge-contracted icosahedron|bidiminished icosahedron]]<br>([2], order 4) |[[File:600-cell verf.svg|120px]]<br>[[W:Icosahedron|icosahedron]]<br>([5,3], order 120) |- align=center !Symmetry |colspan=2|Order 144 (48×3 or 72×2) |[3⁺,4,3]<br>Order 576 (96×6) |[10,2⁺,10]<br>Order 400 (100×4) |[5,3,3]<br>Order 14400 (120×120) |- align=center !Net |[[File:Triicositetradiminished hexacosichoron net.png|100px]] |[[File:Biicositetradiminished hexacosichoron net.png|100px]] |[[File:Snub 24-cell-net.png|100px]] |[[File:Grand antiprism net.png|100px]] |[[File:600-cell net.png|100px]] |- align=center !Ortho<br>H₄ plane | |[[File:bidex ortho-30-gon.png|120px]] |[[File:Snub 24-cell ortho30-gon.png|120px]] |[[File:Grand antiprism ortho-30-gon.png|120px]] |[[File:600-cell graph H4.svg|120px]] |- align=center !Ortho<br>F₄ plane | |[[File:Bidex ortho 12-gon.png|120px]] |[[File:24-cell h01 F4.svg|120px]] |[[File:GrandAntiPrism-2D-F4.svg|120px]] |[[File:600-cell t0 F4.svg|120px]] |} == Related polytopes and honeycombs == The 600-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} {{H4_family}} It is similar to three [[W:Regular 4-polytope|regular 4-polytope]]s: the [[5-cell|5-cell]] {3,3,3}, [[16-cell|16-cell]] {3,3,4} of Euclidean 4-space, and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[W:Tetrahedron|tetrahedral]] cells. {{Tetrahedral cell tessellations}} This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[W:Icosahedron|icosahedron]] vertex figures: {{Icosahedral vertex figure tessellations}} The [[W:regular complex polytope|regular complex polygons]] ₃{5}₃, {{Coxeter–Dynkin diagram|3node_1|5|3node}} and ₅{3}₅, {{Coxeter–Dynkin diagram|5node_1|3|5node}}, in <math>\mathbb{C}^2</math> have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[W:Complex reflection group|complex reflection group]] ₃[5]₃, order 360, and the second has symmetry ₅[3]₅, order 600.{{Sfn|Coxeter|1991|pp=48-49}} {| class="wikitable collapsed collapsible" !colspan=3| Regular complex polytope in orthogonal projection of H₄ Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center |[[File:600-cell graph H4.svg|240px]]<br>{3,3,5}<br>Order 14400 |[[File:Complex polygon 3-5-3.png|240px]]<br>₃{5}₃<br>Order 360 |[[File:Complex polygon 5-3-5.png|240px]]<br>₅{3}₅<br>Order 600 |} == See also == * [[W:600-cell|Wikipedia:600-cell]], the article this article is an expanded version of * [[24-cell|24-cell]], the predecessor 4-polytope on which the 600-cell is based * [[120-cell|120-cell]], the dual 4-polytope to the 600-cell, and its successor * [[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] * [[W:Regular 4-polytope|Regular 4-polytope]] * [[W:Polytope|Polytope]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} *{{Citation | last=Schläfli | first=Ludwig | author-link=W:Ludwig Schläfli |editor-first=Arthur | editor-last=Cayley | editor-link=W:Arthur Cayley | title=An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 | year=1858 | journal=Quarterly Journal of Pure and Applied Mathematics | volume=2 | pages=55–65, 110–120 }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd | isbn= | title-link=W:Regular Complex Polytopes }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | place= | edition=2nd | isbn=978-0-471-01003-6 | url=https://www.wiley.com/en-us/Kaleidoscopes%3A+Selected+Writings+of+H+S+M+Coxeter-p-9780471010036 | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic }} ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }} * {{Cite book|title=The Coxeter Legacy|chapter=Coxeter Theory: The Cognitive Aspects|last=Borovik|first=Alexandre|year=2006|author-link=W:Alexandre Borovik|publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|pages=17–43|isbn=978-0821837221|url=https://www.academia.edu/26091464}} * [[W:John Horton Conway|J.H. Conway]] and [[W:Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965 * [[W:Norman Johnson (mathematician)|N.W. 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It is also known as the '''C₆₀₀''', '''hexacosichoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hexacosihedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and a '''[[W:polytetrahedron|polytetrahedron]]''', being bounded by tetrahedral [[W:Cell (geometry)|cells]]. The 600-cell's boundary is composed of 600 [[W:Tetrahedron|tetrahedral]] [[W:Cell (mathematics)|cells]] with 20 meeting at each vertex.{{Efn|name=vertex icosahedral pyramid}} Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:icosahedron|icosahedron]], since it has five [[W:Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[W:triangle|triangle]]s meeting at every vertex.{{Efn|name=math of dimensional analogy}} Its [[W:dual polytope|dual polytope]] is the [[120-cell|120-cell]]. == Geometry == The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|name=4-polytopes ordered by size and complexity|group=}} It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell|24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[W:Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), "600-cell" column ₀''R/l'' {{=}} 2𝝓/2}} which is the [[W:golden ratio|golden ratio]]. {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === ==== Unit radius Cartesian coordinates ==== The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length {{sfrac|1|φ}} ≈ 0.618 (where φ = {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows: 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates, and 16 vertices of the form: :(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) The remaining 96 vertices are obtained by taking [[W:even permutation|even permutation]]s of :(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ⁻¹|2}}, 0) Note that the first 8 are the vertices of a [[16-cell|16-cell]], the second 16 are the vertices of a [[W:tesseract|tesseract]], and those 24 vertices together are the vertices of a [[24-cell|24-cell]]. The remaining 96 vertices are the vertices of a [[W:snub 24-cell|snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} When interpreted as [[W:#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[W:icosian|icosian]]s. In the 24-cell, there are [[24-cell#Great squares|squares]], [[24-cell#Great hexagons|hexagons]] and [[24-cell#Triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}} In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}} In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]]. The 60 axes and 75 16-cells of the 600-cell constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60₅75₄ to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}} Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs. ==== Hopf spherical coordinates ==== In the 600-cell there are also great circle [[W:pentagon|pentagon]]s and [[W:decagon|decagon]]s (in central planes through ten vertices).{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[W:#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}} Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[W:#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}} By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉_''i'', 𝜂, 𝜉_''j''){{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles. A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two [[W:completely orthogonal|completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<br> : w {{=}} cos 𝜉_''i'' sin 𝜂 : x {{=}} cos 𝜉_''j'' cos 𝜂 : y {{=}} sin 𝜉_''j'' cos 𝜂 : z {{=}} sin 𝜉_''i'' sin 𝜂 The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as: : ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}) where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉_''i'' and 𝜉_''j'' coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}} === Structure === ==== Polyhedral sections ==== The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[W:hypersphere|hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[W:icosahedron|icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[W:dodecahedron|dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[W:icosidodecahedron|icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex}}{{Sfn|Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout|Parker|1998}} These can be seen in the H3 [[W:Coxeter plane|Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} :[[File:600-cell-polyhedral levels.png|640px]] These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron. {| class=wikitable !colspan=2|Concentric Hulls |- |align=center|[[Image:Hulls of H4only-orthonormal.png|360px]] |The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br> <br> 1) two points at the origin<br> 2) two icosahedra<br> 3) two dodecahedra<br> 4) two larger icosahedra<br> 5) and a single icosidodecahedron<br> <br> for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices. |- |} ==== Golden chords ==== [[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]] The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]]. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]] {{see also|W:24-cell#Hypercubic chords|label 1=24-cell § Hypercubic chords}} The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex; see column ''a''}} at eight different [[W:Chord (geometry)|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}} In ascending order of length, they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=The fractional square roots are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <math>\tfrac{1}{\phi} = \phi^{-1}</math> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² ≈ 0.382<br> For example: {{indent|7}}𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618|name=fractional square roots|group=}} Notice that the four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] <big>φ</big> {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to <big>𝜋</big>:<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=W:John Carlos Baez|access-date=10 October 2022}}</ref><br> : {{sfrac|𝜋|5}} = arccos ({{sfrac|φ|2}}) is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord. Reciprocally, in this function discovered by Robert Everest expressing <big>φ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br> : <big>φ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}}) {{sfrac|3𝜋|5}} is the arc length of the <big>φ</big> = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge. The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is <big>φ</big>, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the <big>φ</big> chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}} The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}. The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <big>φ</big>{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}} ==== Boundary envelopes ==== [[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]. The 3D surface made of 600 tetrahedra is visible.]] The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a [[W:snub 24-cell|snub 24-cell]]. Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}} The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: tetrahedra of edge length {{sfrac|1|φ}} ≈ 0.618 instead of octahedra of edge length 1. It encloses the {{radic|1}} edges of the 24-cells, which become invisible interior chords in the 600-cell, like the [[24-cell#Hypercubic chords|{{radic|2}} and {{radic|3}} chords]]. [[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell|24-cell]] performing a [[24-cell#Simple rotations|simple rotation]]. The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.]] Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of {{sfrac|1|φ}}, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[W:Tesseract#Radial equilateral symmetry|radially equilateral]]. Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[W:equilateral triangle|equilateral triangle]]s which all meet at the center.{{Efn|The long radius (center to vertex) of the [[24-cell#geometry|24-cell]] is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[W:Golden triangle (mathematics)|golden triangle]]s which all meet at the center.{{Efn|A [[W:Golden triangle (mathematics)|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ratio|golden ratio]] to the distinct side ''b'': : {{sfrac|a|b}} <nowiki>=</nowiki> φ <nowiki>=</nowiki> {{sfrac|1 + {{radic|5}}|2}} <nowiki>≈</nowiki> 1.618 It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center, and in the regular [[W:Pentagon|pentagon]] by connecting any two adjacent vertices to the vertex opposite them.<br> The vertex angle is: : <nowiki>𝛉 = arccos(</nowiki>{{sfrac|φ|2}}<nowiki>) = </nowiki>{{sfrac|𝜋|5}}<nowiki> = 36°</nowiki> so the base angles are each {{Sfrac|2𝜋|5}} <nowiki>=</nowiki> 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=Golden triangle}} All the [[W:regular polytope|regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[W:Schläfli orthoscheme|orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[W:right triangle|right triangle]]s which meet at the center.{{Efn|The [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is the generalization of the [[W:right triangle|right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[W:golden ratio|golden ratio]] to its edge length; thus its radius is <big>φ</big> if its edge length is 1, and its edge length is {{sfrac|1|φ}} if its radius is 1.}} Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[W:icosidodecahedron|icosidodecahedron]], and the two-dimensional [[W:Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) '''Radially golden''' polytopes are those which can be constructed, with their radii, from [[W:Golden triangle (mathematics)|golden triangles]].{{Efn|name=Golden triangle}} The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an [[W:Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[W:#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} Therefore the successor may be constructed by placing [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[W:cubic pyramid|cubic pyramid]]s. But if we place 24 canonical [[W:octahedral pyramid|octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}} ==== Geodesics ==== The vertex chords of the 600-cell are arranged in [[W:geodesic|geodesic]] [[W:great circle|great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§4 The planes of the 600-cell|pp=437-439}} [[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[W:stereographic projection|stereographic projection]] of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]] The {{radic|0.𝚫}} = 𝚽 edges form 72 flat regular central [[W:decagon|decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}} Just as the [[W:icosidodecahedron|icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 {{radic|0.𝚫}} edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, {{radic|3.𝚽}} apart. As in the decagon and the icosidodecahedron, the edges occur in [[W:Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope. The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[W:Hopf fibration|Hopf fibration]]s, each filling the whole 600-cell. Each [[W:#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} The {{radic|1}} chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}} Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The {{radic|1}} chords join vertices which are two {{radic|0.𝚫}} edges apart. Each {{radic|1}} chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[W:triangular bipyramid|triangular bipyramid]]), and passes through the center of the shared face. As there are 1200 faces, there are 1200 {{radic|1}} chords, in 600 parallel pairs, {{radic|3}} apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell|Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}} The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} The {{radic|1.𝚫}} chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}} The {{radic|1.𝚫}} chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. The {{radic|1.𝚫}} chords join vertices which are two {{radic|0.𝚫}} edges apart on a geodesic great circle. The 720 {{radic|1.𝚫}} chords occur in 360 parallel pairs, {{radic|2.𝚽}} = φ apart. The {{radic|2}} chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex). The {{radic|2}} chords join vertices which are three {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart). There are 600 {{radic|2}} chords, in 300 parallel pairs, {{radic|2}} apart. The 450 great squares (225 [[W:Completely orthogonal|completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} The {{radic|2.𝚽}} = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length {{radic|3.𝚽}}. The {{radic|2.𝚽}} chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 720 distinct {{radic|2.𝚽}} chords, in 360 parallel pairs, {{radic|1.𝚫}} apart. The {{radic|3}} chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 {{radic|3}} chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The {{radic|3}} chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The {{radic|3}} chords join vertices which are four {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart on a geodesic great circle). Each {{radic|3}} chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 {{radic|1}} cubic cells. The 1200 {{radic|3}} chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each {{radic|3}} chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}} There are 1200 {{radic|3}} chords, in 600 parallel pairs, {{radic|1}} apart. The {{radic|3.𝚽}} chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length {{radic|1.𝚫}}, so these are [[W:Golden triangle (mathematics)|golden triangles]]. The {{radic|3.𝚽}} chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 720 distinct {{radic|3.𝚽}} chords, in 360 parallel pairs, {{radic|0.𝚫}} apart. The {{radic|4}} chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell. The {{radic|4}} chords join opposite vertices which are five {{radic|0.𝚫}} edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells. The sum of the squared lengths{{Efn|The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.}} of all these distinct chords of the 600-cell is 14,400 = 120².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[W:completely orthogonal|completely orthogonal]] to a great 30-gon{{Efn|A ''[[W:triacontagon|triacontagon]]'' or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the [[W:Interior angle|interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[W:Equilateral triangle|equilateral triangle]] (60°) and the [[W:Regular pentagon|regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. It is the central axis of a spiral skew 30-gram, the [[W:Petrie polygon|Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}} Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}} All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:icosidodecahedron|icosidodecahedron]]. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens|Seidel|1973}} Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together. Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration. Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles. Vectors in 4-space may be combined by [[W:Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[W:William Rowan Hamilton|Hamilton]].{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}} Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}} Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}} Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}} Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}} Each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one {{radic|4}} long diameter): a great [[W:digon|digon]] plane.{{Efn|In the 24-cell each great square plane is [[W:completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:digon|digon]] plane.|name=digon planes}} Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere. Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell. In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}} ==== Fibrations of great circle polygons ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[W:fiber bundle|fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[W:Hopf fibration|Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}} The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell|Semple|1971|loc=§4. Isoclinic planes in Euclidean space E₄|pp=6-7}} ===== Decagons ===== [[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[W:Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[W:Schläfli double six|Schläfli double six]] configuration 30₂12₅ characteristic of the H₄ polytopes. The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]] The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons. Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[W:icosahedron|icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}} Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}} The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[W:Hopf fibration|Hopf fibration]]s make the 600-cell a [[W:Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30₂12₅. It is called the [[W:Schläfli double six|Schläfli double six]] configuration after [[W:Ludwig Schläfli|Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[W:Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[W:Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3^{2, 2, 1}] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} ===== Hexagons ===== The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration. The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. The Hopf map of this fibration is the [[W:dodecahedron|dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations. The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells. ===== Squares ===== The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the {{radic|2}} tetrahedral cells of the 75 inscribed 16-cells, ''not'' the {{radic|0.𝚫}} tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} The Hopf map of this fibration is the [[W:icosidodecahedron|icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations. ===== Clifford parallel cell rings ===== The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, studied cell rings in the general case of their geometry and [[W:group theory|group theory]], identifying each cell ring as a [[W:polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]].{{Efn|name=orthoscheme ring}} He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:Chiral|chiral]] forms. Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}} Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}} The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}} This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[W:Hopf fibration|Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}} The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}} The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with ''subfibrations''. The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings. The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}} The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope.}} The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[W:Hopf fibration|Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[W:2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}} On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is the [[W:icosahedron|icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration. But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[W:Vertex figure|vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[W:Braid|braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Sfn|Sadoc|Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,{{Efn|name=Clifford parallels}} along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint."}}|name=Hopf fibration base}} of these fibrations on the 2-sphere). Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 {{Background color|red}} faces of the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration''). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}} === Constructions === The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize. ==== Gosset's construction ==== [[W:Thorold Gosset|Thorold Gosset]] discovered the [[W:Semiregular polytope|semiregular 4-polytopes]], including the [[W:Snub 24-cell|snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described [[W:Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[W:Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}} The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}} The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell. Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps. The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[W:Midsphere|midsphere]] to construct its outer [[W:Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell. ==== Cell clusters ==== Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]], or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.{{Efn|name=tetrahedral cell adjacency}} Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional space that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[W:Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point. ===== Icosahedra ===== [[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[W:Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[W:Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[W:Regular icosahedron#Symmetries|snub octahedron]]. This term refers specifically to a [[W:Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}} Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the [[W:Icosahedral pyramid|icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.|alt=|200x200px]] The [[W:Vertex figure|vertex figure]] of the 600-cell is the [[W:Icosahedron|icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical [[W:Icosahedral pyramid|icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|name=radially equilateral icosahedral pyramid}}|name=vertex icosahedral pyramid|group=}} Twenty tetrahedral cells meet at each vertex, forming an [[W:Icosahedral pyramid|icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron. It is remarkable that twenty regular tetrahedra fit inside a regular icosahedral pyramid in 4-space. In 3-space, twenty tetrahedral pyramids fit inside a regular icosahedron around its center but they are ''not'' regular tetrahedra, because the regular icosahedron's radius is not the same as its edge length.{{Efn|In Euclidean 3-space, the icosahedron is not [[W:Cuboctahedron#Radial equilateral symmetry|radially equilateral like the cuboctahedron]]. The icosahedron's radii are shorter than its edge length. But in the [[W:3-sphere|spherical 3-space]] of the 600-cell's surface the center of a regular icosahedron is lifted orthogonally out of its 3-space hyperplane: remarkably, just far enough to make its radii the same length as its edges. As a figure in Euclidean 4-space, this radially equilateral spherical icosahedron is an [[W:Icosahedral pyramid|icosahedral pyramid]]. In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not its center, just one of its vertices. But in curved 3-space the 12 edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that spherical space''. In Euclidean 4-space there are only two radially equilateral figures: 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[W:tesseract#Radial equilateral symmetry|radially equilateral tesseract]].|name=radially equilateral icosahedral pyramid}} The 600-cell has a [[W:Dihedral angle|dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}} An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}} The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[W:Snub 24-cell|snub 24-cell]], which has exactly the same [[W:Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing. The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[W:Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}} Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a [[24-cell#Great hexagons|great circle hexagon]]. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[W:Triangular bipyramid|triangular bipyramid]]) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a {{radic|1}} octahedral cell, but in the larger {{radic|2}} octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]]. The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}} The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[W:Russian dolls|Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}} The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}} The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[W:Jessen's icosahedron|Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron;{{Sfn|Itoh|Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[W:Jitterbug transformation|Jitterbug]] by [[W:Buckminster Fuller|Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}} The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[W:Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca|Al-Mukhaini|Koca|Al Qanobi|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[W:3-sphere|S³]] is similar to the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]] and the [[W:Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[W:Three-dimensional space|R³]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[W:Buckminster Fuller|Buckminster Fuller]].<ref>{{cite journal | last = Verheyen | first = H. F. | doi = 10.1016/0898-1221(89)90160-0 | issue = 1–3 | journal = [[W:Computers and Mathematics with Applications|Computers and Mathematics with Applications]] | mr = 0994201 | pages = 203–250 | title = The complete set of Jitterbug transformers and the analysis of their motion | volume = 17 | year = 1989| doi-access = free }}</ref>}} The tetrahedral cells are face-bonded into [[W:Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[W:Boerdijk-Coxeter helix|Boerdijk-Coxeter helix]]. This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]]. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}} 5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[W:Hopf fibration|Hopf fibration]]. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}} The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other. Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[W:Skew lines|skew edges]] of the tetrahedron. Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[W:Hopf fibration|Hopf fiber bundle]] of 12. Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]]. The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}} Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} ===== Octahedra ===== There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell. Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell. First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[W:Triangular dipyramid|triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length {{radic|1}}. Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length {{radic|1}}. They form a tetrahedron of edge length {{radic|1}}, which is the second section of the 600-cell beginning with a cell.{{Efn|The {{radic|1}} tetrahedron has a volume of 9 {{radic|0.𝚫}} tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the {{radic|1}} tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}} There are 600 of these {{radic|1}} tetrahedral sections in the 600-cell. With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length {{radic|1}}, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}} As partially filled so far (by 17 tetrahedral cells), this {{radic|1}} octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each {{radic|1}} edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}} Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A {{radic|1}} octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same {{radic|1}} octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point {{radic|1}} octahedral section, a 4-point {{radic|1}} tetrahedral section, and a 4-point {{radic|0.𝚫}} tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the {{radic|1}} octahedron surrounds the {{radic|1}} tetrahedron which surrounds the {{radic|0.𝚫}} tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[W:Octahedral pyramid|octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}} Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical {{radic|1}} [[W:Octahedral pyramid|octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter {{radic|0.𝚫}} edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a {{radic|1}} tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with {{radic|0.𝚫}} edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two {{radic|1}} edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three {{radic|0.𝚫}} edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[W:Rectified 5-cell|rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length {{sfrac|1|φ}} ≈ 0.618. ===== Union of two tori ===== There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell. An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines". [[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[W:Clifford torus|Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}} Its opposite edges are identified, forming a [[W:Duocylinder|duocylinder]].]] The [[120-cell|120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]]. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}} Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex-stacked [[W:Icosahedral pyramids|icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This 10-cell ring is shaped like a [[W:Pentagonal antiprism|pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center. This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}} Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}} The surface is the same as that of ten stacked [[W:pentagonal antiprism|pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[W:Triangular bipyramid|triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge. The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}} The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes). Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the [[W:Duocylinder|duocylinder]] and form a [[W:Clifford torus|Clifford torus]].{{Efn|A [[W:Clifford torus|Clifford torus]] is the [[W:Hopf fibration|Hopf fiber bundle]] of a distinct [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[W:3-sphere|3-sphere]], involving all of its points. The [[W:SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[W:Cartesian product|Cartesian product]] of two [[W:Completely orthogonal|completely orthogonal]] [[W:Great circle|great circle]]s. It is a filled [[W:Doughnut|doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[W:4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}} They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]]. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} But how can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[W:Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}} In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[W:Triangular bipyramid|triangular bipyramid]] composed of two tetrahedra. This decomposition of the 600-cell has [[W:Coxeter notation|symmetry]] [10,2⁺,10], order 400, the same symmetry as the [[W:Grand antiprism|grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[W:Pentagonal antiprism|pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}} The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells. ===== Boerdijk–Coxeter helix rings ===== The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[W:Hopf fibration|Hopf fibration]] which fills the entire 600-cell.{{Sfn|Banchoff|1988}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}} Each ring of 30 face-bonded tetrahedra is a cylindrical [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension. {| class="wikitable" width="600" |[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}} |[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}} |[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[W:Skew polygon|skew]] {30/11} [[W:Star polygon|star polygon]] with a [[W:Winding number|winding number]] of 11 called a [[W:Triacontagon#Triacontagram|triacontagram₁₁]], a continuous tight corkscrew [[W:Helix|helix]] bent into a loop of 30 edges (the {{Background color|magenta|magenta}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[W:Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[W:Density (polytope)|characterized]] as the [[W:Petrie polygon|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}} |- |colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''. |} The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three {{Background color|cyan|cyan}} Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}} The 30 {{Background color|magenta|magenta}} edges joining these vertex pairs form a helical [[W:Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[W:Petrie polygon|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[W:Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[W:Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h₁ h₂''.}}|name=Petrie polygon in 30-cell ring}} The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[W:Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell|120-cell]], the 600-cell's [[W:Dual polytope|dual polytope]].{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the 600-cell and its dual the [[120-cell|120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}} The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices. Each orange or yellow edge crosses between two {{Background color|cyan|cyan}} great decagons. Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}} Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[W:Geodesic|geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}} The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between. A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop. The two connected loops forms a single [[W:Möbius loop|Möbius loop]], a skew {15/2} [[W:Pentadecagram|pentadecagram]]. The pentadecagram is not shown in these illustrations (but [[#Decagons and pentadecagrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations. Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}} Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}} They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[W:link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}} Five of these 30-cell [[W:Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[W:Chiral|chiral]] objects; they either spiral clockwise (right) or counterclockwise (left). The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings. Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[W:Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]]. Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration. In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings. Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[W:Hopf link|Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Efn|Sadoc describes the decomposition of the 600-cell into four tori.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} It is the same [[#Decagons|fibration of 12 great decagons and 20 30-cell rings]], seen as a [[#Clifford parallel cell rings|fibration of four completely disjoint 30-cell rings]]{{Efn|name=completely disjoint}} with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we ''can'' discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis, and 6 decagons each. Their axis is a 30-cell ring, not a decagon, and they contain only 3 decagons each. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated ("bumpy") rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring "holes" left to fill in the 600-cell. To do that, fill some of the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both. On the icosahedron Hopf map,{{Efn|name=Hopf fibration base}} the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of "extra flap" like the sixth square flap of the [[W:Cube#Orthogonal projections|net of a cube]] before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus ''every'' 30-cell ring is the center core of a 150-cell ring.}} ==== Radial golden triangles ==== The 600-cell can be constructed radially from 720 [[W:Golden triangle (mathematics)|golden triangle]]s of edge lengths {{radic|0.𝚫}} {{radic|1}} {{radic|1}} which meet at the center of the 4-polytope, each contributing two {{radic|1}} radii and a {{radic|0.𝚫}} edge. They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral {{radic|0.𝚫}} bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular {{radic|0.𝚫}} tetrahedron bases (the cells of the 600-cell). ==== Characteristic orthoscheme ==== {| class="wikitable floatright" !colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\tfrac{1}{\phi} \approx 0.618</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |align=center|<small>164°29′</small> |align=center|<small><math>\pi-2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}} \approx 0.951</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length <small><math>\text{𝒍} = \tfrac{1}{\phi} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center. ==== Reflections ==== The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}} For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram₂ geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}} Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Weyl orbits ==== Another construction method uses [[#Symmetries|quaternions]] and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4): : O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} : O(1000) : V1 : O(0010) : V2 : O(0001) : V3 [[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]] With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell|120-cell]] of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the [[120-cell|120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> === Rotations === The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one [[W:Completely orthogonal|completely orthogonal]] invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]].{{Efn|name=isoclinic geodesic}} The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge-lengths away.{{Efn|name=isoclinic rotation to non-adjacent vertices}} For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex {{radic|1}} (60°) distant, moving {{radic|1/4}} {{=}} 1/2 unit radius in four orthogonal directions.|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles. These '''[[#Rotations on polygram isoclines|isoclines]]''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in [[W:Chiral|chiral]] pairs as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true 5𝝅 circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}} The 600-cell is generated by [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes{{Efn|name=isoclinic invariant planes}} are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:William Kingdon Clifford|Clifford]] displacement is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn||name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane rotates sideways.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} ''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}} ==== Twenty-five 24-cells ==== There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney|Hooker|Johnson|Robinson|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.{{Efn|name=Six orthogonal planes of the Cartesian basis}} The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a {{radic|3}} chord.|name=120° apart}} The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell|Aravind|2009|loc=§3. The 600-cell|pp=2-5}} There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[W:Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}}} Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]]. The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane. There are an infinite number of pairs of [[W:Completely orthogonal|completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}} Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel [[W:Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long. The three Clifford parallel decagons can be seen as the {{Background color|cyan}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]. Each {{Background color|magenta}} edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}} An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}} Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}} The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, and four are left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}|name=rotations to 8 disjoint 24-cells}} The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}} Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}} Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}} With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}} Non-disjoint 24-cells are related by a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane. They are also related by an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[W:Digon|digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}} There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}} As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space. Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}} ''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]]. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}} ''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart. However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}} Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}} (There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}} Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}} In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}. The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}} On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]]. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel. ==== Rotations on polygram isoclines ==== The regular convex 4-polytopes each have their characteristic kind of right (and left) [[W:Isoclinic rotation|isoclinic rotation]], corresponding to their characteristic kind of discrete [[W:Hopf fibration|Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[W:Completely orthogonal|completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.''{{Efn|Consider the statement: ''In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber.'' It can be found in the literature, expressed in the mathematical language of the Hopf fibration,{{Sfn|Kim|Rote|2016|loc= 8 The Construction of Hopf Fibrations|pp=12-16|ps=; see Theorem 13.}} but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting sideways, rotating in parallel with the completely orthogonal great circle.{{Efn|name=isoclinic geodesic}} With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.{{Efn|name=helical geodesic}} Each helical isocline is itself a kind of circle, but it is not a planar great circle of the [[W:Hopf fibration|Hopf fibration]]: it is a special kind of geodesic circle whose circumference is greater than 2𝝅''r'', and it is not pictured explicitly at all by the plain statement we are trying to visualize. We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does ''not'' simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle ''as every great circle itself is moving orthogonally'', flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous ''twisting'' rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once ''in a full isoclinic revolution, every vertex moves more than 360 degrees,'' as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the [[24-cell#Isoclinic rotations|isoclinic rotation of a great hexagon fibration of the 24-cell]], each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle);{{Efn|name=4𝝅 rotation}} but in the [[#Decagons and pentadecagrams|isoclinic rotation of a great decagon fibration of the 600-cell]], each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).}} The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[W:Chain mail|chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell. Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[W:Skew polygon|skew]] ''[[W:Polygram (geometry)|polygrams]]'' (which trace the paths on the [[W:Clifford torus|Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[W:Fiber bundle|fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}} Since isoclinic rotations are [[W:Chiral|chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}} All the fiber bundles are aspects of the same discrete [[W:Hopf fibration|Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations. Cell rings are another expression of the Hopf fibration. Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}} The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells. The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets. Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[W:Loom|loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers. A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[W:Warp and woof|warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}} In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia|Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}} The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[W:Chiral|chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[W:Chiral|chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}} The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[W:Torsion of a curve|torsion]]). Pairs of left-handed and right-handed isoclines run through cell rings of this kind. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}} Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively. Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once. It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers. Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating in "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} Left and right isoclines are different paths that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers. A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation. Each moving vertex travels along an isocline contained within a (moving) cell ring. While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}} The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations. Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines. The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}} A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[W:#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.) In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration. An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices (two or more edge-lengths away){{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex at least two edge-lengths away. In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is exactly two edge-lengths away along one of several great circle geodesic routes: the opposite vertex of a neighboring cell. In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron. |name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind). A left-right pair of isoclinic rotations constitutes a discrete fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}} The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[W:Geodesic|geodesic]]. ==== Decagons and pentadecagrams ==== The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines. The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}} 12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[W:Hopf fibration|Hopf fibration]]. There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}} The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons. The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]]. A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration). The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether. Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices. In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}} In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon. In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a [[W:Hexagram|hexagram]]: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are {{radic|3}} chords of the hexagon instead of {{radic|1}} hexagon edges.{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=4-polytopes ordered by size and complexity}} In the {{radic|1}} [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply {{radic|3}} chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic {6/2} hexagram rotation]] both rotate circles of 6 vertices. The hexagram isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}} The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of hexagrams]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}} In the {{radic|2}} [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its {{radic|2}} edges and its {{radic|4}} diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[W:Winding number|winds]] ''three'' times around the 3-sphere. The invariant central plane completely orthogonal to each {4} great square is another {4} great square {{radic|4}} distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares. The 16-cell's [[W:Dural polytope|dual polytope]] the [[W:8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). In the 8-cell this is a rotation of {{radic|1}} × {{radic|3}} great rectangles, and also a rotation of {{radic|4}} axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]]. In the {{radic|0.𝚫}} [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are {{radic|1}} hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}} The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices. The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices. The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great [[W:Digon|digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell. The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis. The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=4-polytopes ordered by size and complexity}} nested like [[W:Russian dolls|Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}} In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it. Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}} The two non-adjacent vertices are linked by a {{radic|1}} chord of the isocline which is a great hexagon edge (a 24-cell edge). The {{radic|1}} chords of the 30-cell ring (without the {{radic|0.𝚫}} 600-cell edges) form a skew [[W:Triacontagram|triacontagram]]_{{30/4}=2{15/2}} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[W:Pentadecagram|pentadecagram]]₂ isoclines. Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers. Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[W:Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]], '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. (Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.) Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}} Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}} The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations. At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}} Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines. There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell. Each distinct isocline belongs to 5 fiber bundles. {| class="wikitable" width="450" !colspan=4|Three sets of 30-cell ring chords from the same [[W:Orthogonal projection|orthogonal projection]] viewpoint |- ![[W:Pentadecagon#Pentadecagram|Pentadecagram {15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]] |- |colspan=2 align=center|All edges are [[W:Pentadecagram|pentadecagram]] isocline chords of length {{radic|1}}, which are also [[24-cell#Great hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell. |colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length {{radic|1.𝚫}} ≈ 1.176. |- |[[File:Regular_star_polygon_15-2.svg|200px]] |[[File:Regular_star_figure_2(15,2).svg|200px]] |[[File:Regular_star_figure_6(5,1).svg|200px]] |- |valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The {{radic|1}} chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart. |valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The {{radic|1}} chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra. |valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly congruent, each ''acting'' as a left or right isocline in different fibrations. |- |colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. |} Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings, but the two isoclines in each 3-cell ring are directly congruent.{{Efn|The chord-path of an isocline may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Boerdijk–Coxeter helix rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has two helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}} The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one {{radic|0.𝚫}} edge-length apart). The 30 chords joining the isocline's 30 vertices are {{radic|1}} hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell {{radic|0.𝚫}} edges apart on a decagon great circle. {{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram₂ geodesic]] is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions. The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon. The Petrie triacontagon has {{radic|0.𝚫}} edges; the isoclinic pentadecagram₂ has {{radic|1}} edges which join vertices which are two {{radic|0.𝚫}} edges apart. Each {{radic|1}} edge belongs to a different [[#Hexagons|great hexagon]], and successive {{radic|1}} edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}} These isocline chords are both hexa''gon'' edges and penta''gram'' edges. The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other. Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[W:Chessboard|chessboard]].{{Efn|name=isoclinic chessboard}} In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring. ==== Hexagons and hexagrams ==== [[File:Regular_star_figure_2(10,3).svg|thumb|[[W:Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines. Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black {{radic|3}} [[24-cell#Triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring. The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 {{radic|1.𝚫}} chords of each isocline form a skew [[W:Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[W:Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[W:Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]]. Notice the relation between the [[24-cell#Helical hexagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on hexagram isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline because the 600-cell contains multiple 24-cells, and the 600-cell's {{radic|1.𝚫}} isocline chord is shorter than the 24-cell's {{radic|3}} isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on [[24-cell#Helical hexagrams and their 4𝝅 isoclines|hexagrams]], while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference 2𝝅; simple rotations take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[W:20-gon|icosagram]] is a compound of the 24-cell's helical {6/2} hexagram, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}} ==== Squares and octagrams ==== [[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[W:24-gon#Related polygons|{24/5} 24-gram]], with <big>φ</big> {{=}} {{radic|2.𝚽}} edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell ({{radic|1}} edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines. Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines. The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <big>φ</big> {{=}} {{radic|2.𝚽}} chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one {{radic|1}} chord apart, and 5 <big>φ</big> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell. Notice the relations between the [[16-cell#Helical construction|16-cell's rotation of just 2 invariant great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the {{radic|4}} diameter. In the 600-cell vertices are closer together, and its {{radic|2.𝚽}} {{=}} <big>φ</big> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} ! [[W:k-face|''k''-face]]||f_''k''||f₀ || f₁||f₂||f₃||[[W:Vertex figure|''k''-fig]] !Notes |- align=right |H₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|5|node}} ||( ) !f₀ || 120 || 12 || 30 || 20 ||[[W:icosahedron|{3,5}]] || H₄/H₃ = 14400/120 = 120 |- align=right |A₁H₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|5|node}} ||{ } !f₁ || 2 || 720 || 5 || 5 || [[W:pentagon|{5}]] || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |A₂A₁ ||{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node}} ||[[W:equilateral triangle|{3}]] !f₂ || 3 || 3 || 1200 || 2 || { } || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |A₃ ||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}} ||[[W:tetrahedron|{3,3}]] !f₃ || 4 || 6 || 4 || 600|| ( ) || H₄/A₃ = 14400/24 = 600 |} == Symmetries == The [[W:Icosian|icosian]]s are a specific set of Hamiltonian [[W:Quaternion|quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}} The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[W:Rational number|rational number]]s.{{Sfn|Steinbach|1997|p=24}} The finite sums of the 120 [[W:Icosian#Unit icosians|unit icosians]] are called the [[W:Icosian#Icosian ring|icosian ring]]. When interpreted as quaternions,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the 120 vertices of the 600-cell form a [[W:group (mathematics)|group]] under quaternionic multiplication. This group is often called the [[W:Binary icosahedral group|binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[W:Icosahedral group|icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}} It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[W:Invariant subgroup|invariant subgroup]], namely as the subgroup ''2I_L'' of quaternion left-multiplications and as the subgroup ''2I_R'' of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of ''2I_L'' and ''2I_R''; the pair of opposite elements generate the same element of ''RSG''. The [[W:Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''. We have the isomorphism ''RSG ≅ (2I_L × 2I_R) / {Id, -Id}''. The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200. The [[W:Quaternion algebra|quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|2015|p=1|loc="''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}} The binary icosahedral group is [[W:Isomorphic|isomorphic]] to [[W:special linear group|SL(2,5)]]. The full [[W:Symmetry group|symmetry group]] of the 600-cell is the [[W:H4 (mathematics)|Coxeter group H₄]].{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§2 The Labeling of H₄}} This is a [[W:Group (mathematics)|group]] of order 14400. It consists of 7200 [[W:Rotation (mathematics)|rotations]] and 7200 rotation-reflections. The rotations form an [[W:Invariant subgroup|invariant subgroup]] of the full symmetry group. The rotational symmetry group was first described by S.L. van Oss.{{Sfn|Oss|1899||pp=1-18}} The H₄ group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} == Visualization == The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}} One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections. === 2D projections === The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" width=600 |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:600-cell graph H4.svg|200px]]<br>[30]<br>(Red=1) |[[File:600-cell t0 p20.svg|200px]]<br>[20]<br>(Red=1) |[[File:600-cell t0 F4.svg|200px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:600-cell t0 H3.svg|200px]]<br>[10]<br>(Red=1,orange=5,yellow=10) |[[File:600-cell t0 A2.svg|200px]]<br>[6]<br>(Red=1,orange=3,yellow=6) |[[File:600-cell t0.svg|200px]]<br>[4]<br>(Red=1,orange=2,yellow=4) |} === 3D projections === A three-dimensional model of the 600-cell, in the collection of the [[W:Institut Henri Poincaré|Institut Henri Poincaré]], was photographed in 1934–1935 by [[W:Man Ray|Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.<ref>{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&nbsp;58; ''Antony and Cleopatra'', SE-6, p.&nbsp;59; ''mathematical object mo-9'', p.&nbsp;64; ''Merchant of Venice'', SE-9, p.&nbsp;65, and "The Hexacosichoron", Philip Ordning, p.&nbsp;96.</ref> {| class=wikitable !colspan=2|Vertex-first projection |- |[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]] |This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied: * The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown. * The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image. |- !colspan=2|Cell-first projection |- |[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]] |This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied: * The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image. * The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint have been culled for clarity. This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. |} {| class=wikitable !Frame synchronized orthogonal isometric (left) and perspective (right) projections |- |[[File:Cell600Cmp.ogv|640px]] |} == Diminished 600-cells == The [[W:Snub 24-cell|snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell|24-cell]] and taking the [[W:Convex hull|convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[W:Diminishment (geometry)|diminishing]]'' of the 600-cell. The [[W:Grand antiprism|grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} A bi-24-diminished 600-cell, with all [[W:Tridiminished icosahedron|tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells. There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.<ref>{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}</ref> {| class="wikitable collapsible" !colspan=12|Diminished 600-cells |- !Name !Tri-24-diminished 600-cell !Bi-24-diminished 600-cell ![[W:Snub 24-cell|Snub 24-cell]]<br>(24-diminished 600-cell) ![[W:Grand antiprism|Grand antiprism]]<br>(20-diminished 600-cell) !600-cell |- align=center !Vertices |48 |72 |96 |100 |120 |- align=center !Vertex figure<br>(Symmetry) |[[File:Dual tridiminished icosahedron.png|120px]]<br>dual of tridiminished icosahedron<br>([3], order 6) |[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]<br>[[W:Hexahedron|tetragonal antiwedge]]<br>([2]⁺, order 2) |[[File:Snub 24-cell verf.png|120px]]<br>[[W:tridiminished icosahedron|tridiminished icosahedron]]<br>([3], order 6) |[[File:Grand antiprism verf.png|120px]]<br>[[W:Edge-contracted icosahedron|bidiminished icosahedron]]<br>([2], order 4) |[[File:600-cell verf.svg|120px]]<br>[[W:Icosahedron|icosahedron]]<br>([5,3], order 120) |- align=center !Symmetry |colspan=2|Order 144 (48×3 or 72×2) |[3⁺,4,3]<br>Order 576 (96×6) |[10,2⁺,10]<br>Order 400 (100×4) |[5,3,3]<br>Order 14400 (120×120) |- align=center !Net |[[File:Triicositetradiminished hexacosichoron net.png|100px]] |[[File:Biicositetradiminished hexacosichoron net.png|100px]] |[[File:Snub 24-cell-net.png|100px]] |[[File:Grand antiprism net.png|100px]] |[[File:600-cell net.png|100px]] |- align=center !Ortho<br>H₄ plane | |[[File:bidex ortho-30-gon.png|120px]] |[[File:Snub 24-cell ortho30-gon.png|120px]] |[[File:Grand antiprism ortho-30-gon.png|120px]] |[[File:600-cell graph H4.svg|120px]] |- align=center !Ortho<br>F₄ plane | |[[File:Bidex ortho 12-gon.png|120px]] |[[File:24-cell h01 F4.svg|120px]] |[[File:GrandAntiPrism-2D-F4.svg|120px]] |[[File:600-cell t0 F4.svg|120px]] |} == Related polytopes and honeycombs == The 600-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney|Hooker|Johnson|Robinson|2020}} {{H4_family}} It is similar to three [[W:Regular 4-polytope|regular 4-polytope]]s: the [[5-cell|5-cell]] {3,3,3}, [[16-cell|16-cell]] {3,3,4} of Euclidean 4-space, and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[W:Tetrahedron|tetrahedral]] cells. {{Tetrahedral cell tessellations}} This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[W:Icosahedron|icosahedron]] vertex figures: {{Icosahedral vertex figure tessellations}} The [[W:regular complex polytope|regular complex polygons]] ₃{5}₃, {{Coxeter–Dynkin diagram|3node_1|5|3node}} and ₅{3}₅, {{Coxeter–Dynkin diagram|5node_1|3|5node}}, in <math>\mathbb{C}^2</math> have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[W:Complex reflection group|complex reflection group]] ₃[5]₃, order 360, and the second has symmetry ₅[3]₅, order 600.{{Sfn|Coxeter|1991|pp=48-49}} {| class="wikitable collapsed collapsible" !colspan=3| Regular complex polytope in orthogonal projection of H₄ Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center |[[File:600-cell graph H4.svg|240px]]<br>{3,3,5}<br>Order 14400 |[[File:Complex polygon 3-5-3.png|240px]]<br>₃{5}₃<br>Order 360 |[[File:Complex polygon 5-3-5.png|240px]]<br>₅{3}₅<br>Order 600 |} == See also == * [[W:600-cell|Wikipedia:600-cell]], the article this article is an expanded version of * [[24-cell|24-cell]], the predecessor 4-polytope on which the 600-cell is based * [[120-cell|120-cell]], the dual 4-polytope to the 600-cell, and its successor * [[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] * [[W:Regular 4-polytope|Regular 4-polytope]] * [[W:Polytope|Polytope]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} *{{Citation | last=Schläfli | first=Ludwig | author-link=W:Ludwig Schläfli |editor-first=Arthur | editor-last=Cayley | editor-link=W:Arthur Cayley | title=An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 | year=1858 | journal=Quarterly Journal of Pure and Applied Mathematics | volume=2 | pages=55–65, 110–120 }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 |title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=W:Regular Polytopes (book) }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd | isbn= | title-link=W:Regular Complex Polytopes }} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 9iylyp4t4ih8q4p1v0yc5x4w95d6d7t 2691729 2691695 2024-12-13T02:05:44Z Dc.samizdat 2856930 removed duplicate footnote references 2691729 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] rr7b9vkim1x4zafj6ekezq17nd2u6w1 2691733 2691729 2024-12-13T06:39:33Z Dc.samizdat 2856930 /* Rotations */ include Greg Egan's double rotation animation, with caption 2691733 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-cell in double rotation. The 24-vertex 24-cell contains three 8-vertex 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three coordinate systems.]] The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.]] The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] bs98gsturb68kqah2x78j0fr6fjac18 2691755 2691738 2024-12-13T07:57:42Z Dc.samizdat 2856930 /* Planes of rotation */ 2691755 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.]] The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 0gn17natkpircs2h4c556g01r4uqcae 2691756 2691755 2024-12-13T07:59:07Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2691756 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.]] The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.]] The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] a2u7lus9oaadvg0l3jjyzks0f8npa77 0 305363 2691699 2654765 2024-12-12T22:20:26Z Dc.samizdat 2856930 Categories 2691699 wikitext text/x-wiki {{Short description|Four-dimensional analog of the octahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=16-cell<br />(4-orthoplex)| Image_File=Schlegel wireframe 16-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br />(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]<br />4-[[W:Orthoplex|orthoplex]]<br />4-[[W:Demihypercube|demicube]]| Last=[[W:Rectified tesseract|11]]| Index=12| Next=[[W:Truncated tesseract|13]]| Schläfli={3,3,4}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} | Cell_List=16 [[W:Tetrahedron|{3,3}]] [[File:3-simplex t0.svg|25px]]| Face_List=32 [[W:Triangle|{3}]] [[File:2-simplex t0.svg|25px]]| Edge_Count= 24| Vertex_Count= 8| Petrie_Polygon=[[W:Octagon|octagon]]| Coxeter_Group=B₄, [3,3,4], order 384<br />D₄, order 192| Vertex_Figure=[[File:16-cell verf.svg|80px]]<br />[[W:Octahedron|Octahedron]]| Dual=[[W:Tesseract|Tesseract]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]], [[W:Regular polytope|regular]], [[W:Hanner polytope|Hanner polytope]] }} In [[W:Geometry|geometry]], the '''16-cell''' is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician [[W:Ludwig Schläfli|Ludwig Schläfli]] in the mid-19th century.{{Sfn|Coxeter|1973|p=141|loc=§&nbsp;7-x. Historical remarks}} It is also called '''C₁₆''', '''hexadecachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> or '''hexdecahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is the 4-dimesional member of an infinite family of polytopes called [[W:Cross-polytope|cross-polytope]]s, ''orthoplexes'', or ''hyperoctahedrons'' which are analogous to the [[W:Cctahedron|octahedron]] in three dimensions. It is Coxeter's <math>\beta_4</math> polytope.{{Sfn|Coxeter|1973|pp=120=121|loc=§&nbsp;7.2. See illustration Fig 7.2<small>B</small>}} The [[W:Dual polytope|dual polytope]] is the [[W:Tesseract|tesseract]] (4-[[W:Hypercube|cube]]), which it can be combined with to form [[W:Compound of tesseract and 16-cell|a compound figure]]. The cells of the 16-cell are dual to the 16 vertices of the tesseract. == Geometry == The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 16-cell is the 8-point 4-polytope: second in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}} Each of its 4 successor convex regular 4-polytopes can be constructed as the [[W:Convex hull|convex hull]] of a [[W:Polytope compound|polytope compound]] of multiple 16-cells: the 16-vertex [[W:Tesseract|tesseract]] as a compound of two 16-cells, the 24-vertex [[24-cell]] as a compound of three 16-cells, the 120-vertex [[600-cell]] as a compound of fifteen 16-cells, and the 600-vertex [[120-cell]] as a compound of seventy-five 16-cells.{{Efn|There are 2 and only 2 16-cells inscribed in the 8-cell (tesseract), 3 and only 3 16-cells inscribed in the 24-cell, 75 distinct 16-cells (but only 15 disjoint 16-cells) inscribed in the 600-cell, and 675 distinct 16-cells (but only 75 disjoint 16-cells) inscribed in the 120-cell.}} {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === {| class="wikitable floatright" !colspan=2|Disjoint squares |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''xy'' plane |- |( 0, 1, 0, 0)||( 0, 0,-1, 0) |- |( 0, 0, 1, 0)||( 0,-1, 0, 0) |} |- | {| class="wikitable" style="white-space:nowrap;" !colspan=2|''wz'' plane |- |( 1, 0, 0, 0)||( 0, 0, 0,-1) |- |( 0, 0, 0, 1)||(-1, 0, 0, 0) |} |}The 16-cell is the 4-dimensional [[W:Cross polytope|cross polytope (4-orthoplex)]], which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}. The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in the 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices).{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} The 16-cell constitutes an [[W:Orthonormal basis|orthonormal ''basis'']] for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes. === Structure === The [[W:Schläfli symbol|Schläfli symbol]] of the 16-cell is {3,3,4}, indicating that its cells are [[W:Regular tetrahedron|regular tetrahedra]] {3,3} and its [[W:Vertex figure|vertex figure]] is a [[W:Regular octahedron|regular octahedron]] {3,4}. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its [[W:Edge figure|edge figure]] is a square. There are 4 tetrahedra and 4 triangles meeting at every edge. The 16-cell is [[W:Totally bounded|bounded]] by 16 [[W:Cell (mathematics)|cells]], all of which are regular [[W:Tetrahedron|tetrahedra]].{{Efn|The boundary surface of a 16-cell is a finite 3-dimensional space consisting of 16 tetrahedra arranged face-to-face (four around one). It is a closed, tightly curved (non-Euclidean) 3-space, within which we can move straight through 4 tetrahedra in any direction and arrive back in the tetrahedron where we started. We can visualize moving around inside this tetrahedral [[W:Jungle gym|jungle gym]], climbing from one tetrahedron into another on its 24 struts (its edges), and never being able to get out (or see out) of the 16 tetrahedra no matter what direction we go (or look). We are always on (or in) the ''surface'' of the 16-cell, never inside the 16-cell itself (nor outside it). We can see that the 6 edges around each vertex radiate symmetrically in 3 dimensions and form an orthogonal 3-axis cross, just as the radii of an octahedron do (so we say the vertex figure of the 16-cell is the octahedron).{{Efn|name=octahedral pyramid}}}} It has 32 [[W:Triangle (geometry)|triangular]] [[W:Face (geometry)|faces]], 24 [[W:Edge (geometry)|edges]], and 8 [[W:Vertex (geometry)|vertices]]. The 24 edges bound 6 [[W:Orthogonal|orthogonal]] central squares lying on [[W:Great circle|great circles]] in the 6 coordinate planes (3 pairs of completely orthogonal{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O. A and B are perpendicular ''and'' [[#Octahedral dipyramid|Clifford parallel]].{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}} great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the [[W:Apex (geometry)|apex]] of a canonical [[W:Octahedral pyramid|octahedral pyramid]].{{Efn|Each vertex in the 16-cell is the apex of an [[W:Octahedral pyramid|octahedral pyramid]], the base of which is the octahedron formed by the 6 other vertices to which the apex is connected by edges. The 16-cell can be deconstructed (four different ways) into two octahedral pyramids by cutting it in half through one of its four octahedral central hyperplanes. Looked at from inside the curved 3 dimensional volume of its boundary surface of 16 face-bonded tetrahedra, the 16-cell's vertex figure is an octahedron. In 4 dimensions, the vertex octahedron is actually an octahedral pyramid. The apex of the octahedral pyramid (the vertex where the 6 edges meet) is not actually at the center of the octahedron: it is displaced radially outwards in the fourth dimension, out of the hyperplane defined by the octahedron's 6 vertices. The 6 edges around the vertex make an orthogonal 3-axis cross in 3 dimensions (and in the [[W:Octahedral pyramid|3-dimensional projection of the 4-pyramid]]), but the 3 lines are actually bent 90 degrees in the fourth dimension where they meet in an apex.|name=octahedral pyramid}} The 6 orthogonal central planes of the 16-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:Octahedron|octahedron]] with 3 orthogonal great squares. === Rotations === {| class="wikitable" width=480 |- align=center valign=top |rowspan=2|[[File:16-cell.gif]]<br />A 3D projection of a 16-cell performing a [[W:SO(4)#Simple rotations|simple rotation]] |[[File:16-cell-orig.gif]]<br />A 3D projection of a 16-cell performing a [[W:SO(4)#Double rotations|double rotation]] |} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§&nbsp;5. Four-Dimensional Rotations}} The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).{{Efn|name=six orthogonal planes of the Cartesian basis}} Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the ''xy'' plane) and another angle of rotation in the completely orthogonal great square plane (the ''wz'' plane).{{Efn|Each great square vertex is {{radic|2}} distant from two of the square's other vertices, and {{radic|4}} distant from its opposite vertex. The other four vertices of the 16-cell (also {{radic|2}} distant) are the vertices of the square's completely orthogonal square.{{Efn|name=Clifford parallel great squares}} Each 16-cell vertex is a vertex of ''three'' orthogonal great squares which intersect there. Each of them has a different ''completely'' orthogonal square. Thus there are three great squares completely orthogonal to each vertex: squares that the vertex is not part of.{{Efn|The three ''incompletely'' orthogonal great squares which intersect at each vertex of the 16-cell form the vertex's octahedral [[W:Vertex figure|vertex figure]].{{Efn|name=octahedral pyramid}} Any two of them, together with the completely orthogonal square of the third, also form an octahedron: a central octahedral hyperplane.{{Efn|Three great squares meet at each vertex (and at its opposite vertex) in the 16-cell. Each of them has a different completely orthogonal square.{{Efn|name=completely orthogonal planes}} Thus there are three great squares completely orthogonal to each vertex and its opposite vertex (each axis). They form an octahedron (a central hyperplane). Every axis line in the 16-cell is completely orthogonal to a central octahedron hyperplane, as every great square plane is completely orthogonal to another great square plane.{{Efn|name=six orthogonal planes of the Cartesian basis}} The axis and the octahedron intersect only at one point (the center of the 16-cell), as each pair of completely orthogonal great squares intersects only at one point (the center of the 16-cell). Each central octahedron is also the octahedral vertex figure of two of the eight vertices: the two on its completely orthogonal axis.|name=octahedral hyperplanes}} In the 16-cell, each octahedral vertex figure is also a central octahedral hyperplane.|name=completely orthogonal great squares}}|name=vertex and central octahedra}} Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.{{Efn|Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are [[W:Clifford parallel|Clifford parallel]].{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§&nbsp;3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=7-10|loc=§&nbsp;6. Angles between two Planes in 4-Space}} In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all {{radic|2}} apart, so these squares are Clifford parallel polygons.{{Efn|name=completely orthogonal Clifford parallels are special}} Note that only the vertices of the great squares (the points on the great circle) are {{radic|2}} apart; points on the edges of the squares (on chords of the circle) are closer together.|name=Clifford parallels}} They are {{radic|2}} apart at each pair of nearest vertices (and in the 16-cell ''all'' the pairs except antipodal pairs are nearest). The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.{{Efn|name=six orthogonal planes of the Cartesian basis}} Because they are perpendicular and share a common center, the two squares are obviously not parallel and separate in the usual way of parallel squares in 3 dimensions; rather they are connected like adjacent square links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallel great squares}} In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]], in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.) In a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] takes place.{{Efn|In an isoclinic rotation, all 6 orthogonal planes are displaced in two orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. An isoclinic displacement (also known as a [[W:William Kingdon Clifford|Clifford]] displacement) is 4-dimensionally diagonal. Points are displaced an equal distance in four orthogonal directions at once, and displaced a total [[W:Pythagorean distance#Higher dimensions|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices of a regular 4-polytope are displaced to a vertex at least two edge lengths away. For example, when the unit-radius 16-cell rotates isoclinically 90° in a great square invariant plane, it also rotates 90° in the completely orthogonal great square invariant plane.{{Efn||name=six orthogonal planes of the Cartesian basis}} The great square plane also tilts sideways 90° to occupy its completely orthogonal plane. (By isoclinic symmetry, ''every'' great square rotates 90° ''and'' tilts sideways 90° into its completely orthogonal plane.) Each vertex (in every great square) is displaced to its antipodal vertex, at a distance of {{radic|1}} in each of four orthogonal directions, a total distance of {{radic|4}}.{{Efn|Opposite vertices in a unit-radius 4-polytope correspond to the opposite vertices of an 8-cell hypercube (tesseract). The long diagonal of this [[W:Tesseract#Radial equilateral symmetry|radially equilateral 4-cube]] is {{radic|4}}. In a 90° isoclinic rotation each vertex of the 16-cell is displaced to its antipodal vertex, traveling along a helical geodesic arc of length 𝝅 (180°), to a vertex {{radic|4}} away along the long diameter of the unit-radius 4-polytope (16-cell or tesseract), the same total displacement as if it had been displaced {{radic|1}} four times by traveling along a path of four successive orthogonal edges of the tesseract.|name=long diagonal of the 4-cube}} The original and displaced vertex are two edge lengths apart by three{{Efn|There are six different two-edge paths connecting a pair of antipodal vertices along the edges of a great square. The left isoclinic rotation runs diagonally between three of them, and the right isoclinic rotation runs diagonally between the other three. These diagonals are the straight lines (geodesics) connecting opposite vertices of face-bonded tetrahedral cells in the left-handed [[#Helical construction|eight-cell ring]] and the right-handed eight-cell ring, respectively.}} different paths along two edges of a great square. But the ''isocline'' (the helical arc the vertex follows during the isoclinic rotation) does not run along edges: it runs ''between'' these different edge-paths diagonally, on a geodesic (shortest arc) between the original and displaced vertices.{{Efn|name=isocline}} This isoclinic geodesic arc is not a segment of an ordinary great circle; it does not lie in the plane of any great square. It is a helical 180° arc that bends in a circle in two completely orthogonal planes at once. This [[W:Möbius loop|Möbius circle]] does not lie in any plane or intersect any vertices between the original and the displaced vertex.{{Efn|name=Möbius circle}}|name=isoclinic rotation}} In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 16-cell, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel){{Efn|name=Clifford parallels}} plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great squares are {{radic|4}} (180°) apart; the great squares (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them. If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great square returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 16-cell (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}} === Constructions === ==== Octahedral dipyramid ==== {|class="wikitable floatright" !Octahedron <math>\beta_3</math> !16-cell <math>\beta_4</math> |- |[[File:3-cube t2.svg|160px]] |[[File:4-demicube t0 D4.svg|160px]] |- |colspan=2|Orthogonal projections to skew hexagon hyperplane |} The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the [[W:Octahedron|octahedron]]. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its [[W:Petrie polygon|Petrie polygon]] is the [[W:Hexagon|hexagon]]). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two [[W:Octahedral pyramid|octahedral pyramid]]s on a shared octahedron base that lies in the 16-cell's central hyperplane.{{Sfn|Coxeter|1973|p=121|loc=§&nbsp;7.21. See illustration Fig 7.2<small>B</small>|ps=: "<math>\beta_4</math> is a four-dimensional dipyramid based on <math>\beta_3</math> (with its two apices in opposite directions along the fourth dimension)."}} [[File:stereographic_polytope_16cell_colour.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 16-cell's 6 orthogonal central squares onto their great circles. Each circle is divided into 4 arc-edges at the intersections where 3 circles cross perpendicularly. Notice that each circle has one Clifford parallel circle that it does ''not'' intersect. Those two circles pass through each other like adjacent links in a chain.]]The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with ''two'' of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and ''three more squares'' (which appear edge-on as the 3 ''diameters'' of the hexagon in the projection), and three more octahedra.{{Efn|name=octahedral hyperplanes}} Something unprecedented has also been created. Notice that each square no longer intersects with ''all'' of the other squares: it does intersect with four of them (with ''three'' of the squares crossing at each vertex now), but each square has ''one'' other square with which it shares ''no'' vertices: it is not directly connected to that square at all. These two ''separate'' perpendicular squares (there are three pairs of them) are like the opposite edges of a [[W:Tetrahedron|tetrahedron]]: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of '''''Clifford parallel planes''''', and the 16-cell is the simplest regular polytope in which they occur. [[W:William Kingdon Clifford|Clifford]] parallelism{{Efn|name=Clifford parallels}} of objects of more than one dimension (more than just curved ''lines'') emerges here and occurs in all the subsequent 4-dimensional regular polytopes, where it can be seen as the defining relationship ''among'' disjoint concentric regular 4-polytopes and their corresponding parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.{{Sfn|Tyrrell|Semple|1971}} For example, as noted [[#Geometry|above]] all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]]. ==== Tetrahedral constructions ==== {| class="wikitable" width=480 |- align=center valign=top |[[File:16-cell net.png|180px|]] |[[File:16-cell nets.png|180px]] |} The 16-cell has two [[W:Wythoff construction|Wythoff construction]]s from regular tetrahedra, a regular form and alternated form, shown here as [[W:Net (polyhedron)|nets]], the second represented by tetrahedral cells of two alternating colors. The alternated form is a [[#Symmetry constructions|lower symmetry construction]] of the 16-cell called the [[W:Demitesseract|demitesseract]]. Wythoff's construction replicates the 16-cell's [[5-cell#Orthoschemes|characteristic 5-cell]] in a [[W:Kaleidoscope|kaleidoscope]] of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facets]] (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} There are three regular 4-polytopes with tetrahedral cells: the [[5-cell]], the 16-cell, and the [[600-cell]]. Although all are bounded by ''regular'' tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different [[5-cell#Isometries|tetrahedral pyramids]], all based on the same characteristic ''irregular'' tetrahedron. They share the same [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron]] (3-orthoscheme) and characteristic [[W:Right triangle|right triangle]] (2-orthoscheme) because they have the same kind of cell.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§&nbsp;7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} {| class="wikitable floatright" !colspan=6|Characteristics of the 16-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "16-cell, 𝛽₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§&nbsp;7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{2} \approx 1.414</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center|<small>60″</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45″</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30″</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center| |align=center| |align=center| |align=center| |} The '''characteristic 5-cell of the regular 16-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|4|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center. The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 16-cell has unit radius edge and edge length 𝒍 = <small><math>\sqrt{2}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small> (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center. ==== Helical construction ==== [[File:Eight face-bonded tetrahedra.jpg|thumb|A 4-dimensional ring of 8 face-bonded tetrahedra, seen in the [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]], bounded by three eight-edge circular paths of different colors, cut and laid out flat in 3-dimensional space. It contains an ''isocline'' axis (not shown), a helical circle of circumference 4𝝅 that twists through all four dimensions and visits all 8 vertices.{{Efn|name=isocline}} The two blue-blue-yellow triangles at either end of the cut ring are the same object.]] [[File:16-cell 8-ring net4.png|thumb|Net and orthogonal projection]] A 16-cell can be constructed (three different ways) from two [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]es of eight chained tetrahedra, each bent in the fourth dimension into a ring.{{Sfn|Coxeter|1970|loc=Table 2: Reflexible honeycombs and their groups|p=45|ps=; Honeycomb [3,3,4]₄ is a tiling of the 3-sphere by 2 rings of 8 tetrahedral cells.}}{{Sfn|Banchoff|2013}} The two circular helixes spiral around each other, nest into each other and pass through each other forming a [[W:Hopf link|Hopf link]]. The 16 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex. The purple edges represent the [[W:Petrie polygon|Petrie polygon]] of the 16-cell. The eight-cell ring of tetrahedra contains three [[W:Octagram|octagram]]s of different colors, eight-edge circular paths that wind twice around the 16-cell on every third vertex of the octagram. The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a [[W:Möbius strip|Möbius strip]]. Thus the 16-cell can be decomposed into two cell-disjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The left-handed and right-handed cell rings fit together, nesting into each other and entirely filling the 16-cell, even though they are of opposite chirality. This decomposition can be seen in a 4-4 [[W:Duoantiprism|duoantiprism]] construction of the 16-cell: {{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} or {{Coxeter–Dynkin diagram|node|4|node_h|2x|node_h|4|node}}, [[W:Schläfli symbol|Schläfli symbol]] {2}⨂{2} or s{2}s{2}, [[W:Coxeter notation|symmetry]] [4,2⁺,4], order 64. Three eight-edge paths (of different colors) spiral along each eight-cell ring, making 90° angles at each vertex. (In the Boerdijk–Coxeter helix before it is bent into a ring, the angles in different paths vary, but are not 90°.) Three paths (with three different colors and apparent angles) pass through each vertex. When the helix is bent into a ring, the segments of each eight-edge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 4𝝅, and one edge wide.{{Efn|name=Möbius circle}} The six four-edge halves of the three eight-edge paths each make four 90° angles, but they are ''not'' the six orthogonal great squares: they are open-ended squares, four-edge 360° helices whose open ends are [[W:Antipodal point|antipodal]] vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined end-to-end in pairs of the same [[W:Chirality|chirality]], the six four-edge paths make three eight-edge Möbius loops, [[W:Helix|helical]] octagrams. Each octagram is both a [[W:Petrie polygon|Petrie polygon]] of the 16-cell, and the helical track along which all eight vertices rotate together, in one of the 16-cell's distinct isoclinic [[#Rotations|rotations]].{{Efn|The 16-cell can be constructed from two cell-disjoint eight-cell rings in three different ways; it has three orientations of its pair of rings. Each orientation "contains" a distinct left-right pair of isoclinic rotations, and also a pair of completely orthogonal great squares (Clifford parallel fibers), so each orientation is a discrete [[W:Hopf fibration|fibration]] of the 16-cell. Each eight-cell ring contains three axial octagrams which have different orientations (they exchange roles) in the three discrete fibrations and six distinct isoclinic rotations (three left and three right) through the cell rings. Three octagrams (of different colors) can be seen in the illustration of a single cell ring, one in the role of Petrie polygon, one as the right isocline, and one as the left isocline. Because each octagram plays three roles, there are exactly six distinct isoclines in the 16-cell, not 18.|name=only one disjoint pair of eight-cell rings}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at the same [[W:Skew polygon|skew]] [[W:Octagram|octagram]]{{Efn|All five views are the same orthogonal projection of the 16-cell into the same plane (a circular cross-section of the eight-cell ring cylinder), looking along the central axis of the cut ring cylinder pictured above, from one end of the cylinder. The only difference is which {{radic|2}} edges and {{radic|4}} chords are ''omitted'' for focus. The different colors of {{radic|2}} edges appear to be of different lengths because they are oblique to the viewer at different angles. Vertices are numbered 1 (top) through 8 in counterclockwise order.}} |- ![[#Rotations|Edge path]] ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]]{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁''}} !16-cell ![[W:Hopf fibration|Discrete fibration]] ![[#Coordinates|Diameter chords]] |- ![[W:Octagram|Octagram]]_{8/3}{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₂''}} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Octagram]]_{8/1} ![[W:Coxeter element#Coxeter plane|Coxeter plane]] [[W:B4 polytope|B₄]] ![[W:Octagram#Star polygon compounds|Octagram]]_{8/2}=2{4} ![[W:Octagram#Star polygon compounds|Octagram]]_{8/4}=4{2} |- |align=center|[[File:16-cell skew octagram (8-3).png|120px]] |align=center|[[File:16-cell skew octagram (8).png|120px]] |align=center|[[File:16-cell skew octagram.png|120px]] |align=center|[[File:16-cell skew octagram 2(4).png|120px]] |align=center|[[File:16-cell skew octagram 4(2).png|120px]] |- |The eight {{radic|2}} chords of the edge-path of an isocline.{{Efn|name=isocline curve}} |Skew [[W:Octagon|octagon]] of eight {{radic|2}} edges. The 16-cell has 3 of these 8-vertex circuits. |All 24 {{radic|2}} edges and the four {{radic|4}} orthogonal axes. |Two completely orthogonal (disjoint) great squares of {{radic|2}} edges.{{Efn|name=Clifford parallel great squares}} |The four {{radic|4}} chords of an isocline. Every fourth isocline vertex is joined to its antipodal vertex by a 16-cell axis.{{Efn|Each isocline has the eight {{radic|2}} chords of its edge-path, and also four {{radic|4}} diameter chords that connect every fourth vertex on the hexagram_{8/3}. Antipodal vertices also have a twisted path of four mutually orthogonal {{radic|2}} edges connecting them. Between antipodal vertices, the isocline curves smoothly around in a helix over the {{radic|2}} chords of its edge-path, hitting ''three'' intervening vertices. Each {{radic|2}} edge is an edge of a great square, that is completely orthogonal to another great square, in which the {{radic|4}} chord is a diagonal.|name=isocline curve}} |} Each eight-edge helix is a [[W:Skew polygon|skew]] [[W:Octagram|octagram]]_{8/3} that [[W:Winding number|winds three times]] around the 16-cell and visits every vertex before closing into a loop. Its eight {{radic|2}} edges are chords of an ''isocline'', a helical arc on which the 8 vertices circle during an isoclinic rotation.{{Efn|An isocline is a circle of special kind corresponding to a pair of [[W:Villarceau circle|Villarceau circle]]s linked in a [[W:Möbius loop|Möbius loop]]. It curves through four dimensions instead of just two. All ordinary circles have a 2𝝅 circumference, but the 16-cell's isocline is a circle with an 4𝝅 circumference (over eight 90° chords). An isocline is a circle that does not lie in a plane, but to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''circle'' for an ordinary circle in the plane.|name=Möbius circle}} All eight 16-cell vertices are {{radic|2}} apart except for opposite (antipodal) vertices, which are {{radic|4}} apart. A vertex moving on the isocline visits three other vertices that are {{radic|2}} apart before reaching the fourth vertex that is {{radic|4}} away.{{Efn|In the 16-cell, two antipodal vertices are opposite vertices of two face-bonded tetrahedral cells. The two antipodal vertices are connected by (three different) two-edge great circle paths along edges of the tetrahedral cells, by various three-edge paths, and by four-edge paths on isoclines and Petrie polygons. {{Efn|name=Möbius circle}}|name=isocline}} The eight-cell ring is [[W:Chiral|chiral]]: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eight-cell ring twists. Each isocline visits all eight vertices of the 16-cell.{{Efn|In the 16-cell each ''single'' isocline winds through all 8 vertices: an entire [[W:Hopf fibration|fibration]] of two completely orthogonal great squares.{{Efn|name=completely orthogonal Clifford parallels are special}} The 5-cell and the 16-cell are the only regular 4-polytopes where each discrete fibration has just one isocline fiber.{{Efn|Except in the 5-cell and 16-cell,{{Efn|name=two special cases}} a pair of left and right isocline circles have disjoint vertices: the left and right isocline helices are non-intersecting parallels but counter-rotating, forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect).|name=counter-rotating double helix}}|name=each 16-cell isocline reaches all 8 vertices}} Each eight-cell ring contains half of the 16 cells, but all 8 vertices; the two rings share the vertices, as they nest into each other and fit together. They also share the 24 edges, though left and right octagram helices are different eight-edge paths.{{Efn|The left and right isoclines intersect each other at every vertex. They are different sequences of the same set of 8 vertices. With respect only to the set of 4 vertex pairs which are {{radic|2}} apart, they can be considered to be Clifford parallel. With respect only to the set of 4 vertex pairs which are {{radic|4}} apart, they can be considered to be completely orthogonal.{{Efn|name=completely orthogonal Clifford parallels are special}}}} Because there are three pairs of completely orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} there are three congruent ways to compose a 16-cell from two eight-cell rings. The 16-cell contains three left-right pairs of eight-cell rings in different orientations, with each cell ring containing its axial isocline.{{Efn|name=only one disjoint pair of eight-cell rings}} Each left-right pair of isoclines is the track of a left-right pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation.{{Efn|name=Clifford parallel great squares}} At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16-cell axis chord.{{Efn|This is atypical for isoclinic rotations generally; normally both the left and right isoclines do not occur at the same vertex: there are two disjoint sets of vertices reachable only by the left or right rotation respectively.{{Efn|name=counter-rotating double helix}} The left and right isoclines of the 16-cell form a very special double helix: unusual not just because it is circular, but because its different left and right helices twist around each other through the ''same set'' of antipodal vertices,{{Efn|name=each 16-cell isocline reaches all 8 vertices}} not through the two ''disjoint subsets'' of antipodal vertices, as the isocline pairs do in most isoclinic rotations found in nature.{{Efn|For another example of the left and right isoclines of a rotation visiting the same set of vertices, see the [[5-cell#Geodesics and rotations|characteristic isoclinic rotation of the 5-cell]]. Although in these two special cases left and right isoclines of the same rotation visit the same set of vertices, they still take very different rotational paths because they visit the same vertices in different sequences.|name=two special cases}} Isoclinic rotations in completely orthogonal invariant planes are special.{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal. There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]]. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'' unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|pp=7-8|loc=§&nbsp;6 Angles between two Planes in 4-Space|ps=; Left and Right Pairs of Isoclinic Planes.}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. Because planes separated by a 90° isoclinic rotation are 180° apart, the plane to the ''left'' and the plane to the ''right'' are the same plane.{{Efn|name=exchange of completely orthogonal planes}}|name=completely orthogonal Clifford parallels are special}} To see ''how'' and ''why'' they are special, visualize two completely orthogonal invariant planes of rotation, each rotating by some rotation angle ''and'' tilting sideways by the same rotation angle into a different plane entirely.{{Efn|name=isoclinic rotation}} ''Only when the rotation angle is 90°,'' that different plane in which the tilting invariant plane lands will be the completely orthogonal invariant plane itself. The destination plane of the rotation ''is'' the completely orthogonal invariant plane. The 90° isoclinic rotation is the only rotation which takes the completely orthogonal invariant planes to each other.{{Efn|name=exchange of completely orthogonal planes}} This reciprocity is the reason both left and right rotations go to the same place.}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}8 & 6 & 12 & 8 \\ 2 & 24 & 4 & 4 \\ 3 & 3 & 32 & 2 \\ 4 & 6 & 4 & 16 \end{matrix}\end{bmatrix}</math> == Tessellations == One can [[W:Tessellation|tessellate]] 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 16-cells. This is called the [[W:16-cell honeycomb|16-cell honeycomb]] and has [[W:Schläfli symbol|Schläfli symbol]] {3,3,4,3}. Hence, the 16-cell has a [[W:Dihedral angle|dihedral angle]] of 120°.{{sfn|Coxeter|1973|p=293}} Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation. The dual tessellation, the [[W:24-cell honeycomb|24-cell honeycomb]], {3,4,3,3}, is made of regular [[24-cell]]s. Together with the [[W:Tesseractic honeycomb|tesseractic honeycomb]] {4,3,3,4} these are the only three [[W:List of regular polytopes#Tessellations of Euclidean 4-space|regular tessellations]] of '''R'''⁴. == Projections == {{B4 Coxeter plane graphs|t3|150}} [[File:Orthogonal projection envelopes 16-cell.png|thumb|Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)]] The cell-first parallel projection of the 16-cell into 3-space has a [[W:cube|cubical]] envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope. The cell-first perspective projection of the 16-cell into 3-space has a [[W:triakis tetrahedron|triakis tetrahedral]] envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection. The vertex-first parallel [[W:Graphical projection|projection]] of the 16-cell into 3-space has an [[W:octahedron|octahedral]] [[W:projection envelope|envelope]]. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron. Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a [[W:hexagonal bipyramid]]al envelope. == 4 sphere Venn diagram == A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a [[W:Venn diagram|Venn diagram]] of 4 sets) are [[W:topology|topologically]] equivalent. {| |- | {{multiple image | align = left | total_width = 700 | image1 = 4 spheres, cell 00, solid.png | image2 = 4 spheres, weight 1, solid.png | image3 = 4 spheres, weight 2, solid.png | image4 = 4 spheres, weight 3, solid.png | image5 = 4 spheres, cell 15, solid.png | footer = The 16 cells ordered by number of intersecting spheres (from 0 to 4) &nbsp;&nbsp;&nbsp; <small>(see all [[commons:Category:Venn diagrams rgby; single cells|cells]] and [[v:Tesseract and 16-cell faces|''k''-faces]])</small> }} | {{multiple image | align = right | total_width = 290 | image1 = 4 spheres as rings, vertical.png | image2 = Stereographic polytope 16cell.png | footer = 4 sphere Venn diagram and 16-cell projection in the same orientation }} |} == Symmetry constructions == The 16-cell's [[W:Coxeter group|symmetry group]] is denoted [[W:B4 polytope|B₄]]. There is a lower symmetry form of the ''16-cell'', called a '''demitesseract''' or '''4-demicube''', a member of the [[W:Demihypercube|demihypercube]] family, and represented by h{4,3,3}, and [[W:Coxeter diagram|Coxeter diagram]]s {{Coxeter–Dynkin diagram|node_h1|4|node|3|node|3|node}} or {{Coxeter–Dynkin diagram|nodes_10ru|split2|node|3|node}}. It can be drawn bicolored with alternating [[W:tetrahedron|tetrahedral]] cells. It can also be seen in lower symmetry form as a '''tetrahedral antiprism''', constructed by 2 parallel [[W:tetrahedron|tetrahedra]] in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: {{Coxeter–Dynkin diagram|node_h|2x|node_h|4|node|3|node}}. It can also be seen as a snub 4-[[W:Orthotope|orthotope]], represented by s{2^1,1,1}, and Coxeter diagram: {{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} or {{Coxeter–Dynkin diagram|node_h|2x|node_h|split1-22|nodes_hh}}. With the [[W:Tesseract|tesseract]] constructed as a 4-4 [[W:Duoprism|duoprism]], the 16-cell can be seen as its dual, a 4-4 [[W:Duopyramid|duopyramid]]. {| class=wikitable !Name ![[W:Coxeter diagram|Coxeter diagram]] ![[W:Schläfli symbol|Schläfli symbol]] ![[W:Coxeter notation|Coxeter notation]] !Order ![[W:Vertex figure|Vertex figure]] |- align=center !Regular 16-cell |{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} |{3,3,4} |[3,3,4]||384 |{{Coxeter–Dynkin diagram|node_1|3|node|4|node}} |- align=center !Demitesseract<br />[[W:Quasiregular polytope|Quasiregular]] 16-cell |{{Coxeter–Dynkin diagram|nodes_10ru|split2|node|3|node}} = {{Coxeter–Dynkin diagram|node_h1|4|node|3|node|3|node}}<br />{{Coxeter–Dynkin diagram|node_1|3|node|split1|nodes}} = {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node_h0}} |h{4,3,3}<br />{3,3^1,1} |[3^1,1,1] = [1⁺,4,3,3]||192 |{{Coxeter–Dynkin diagram|node|3|node_1|3|node}} |- align=center !Alternated 4-4 [[W:Duoprism|duoprism]] |{{Coxeter–Dynkin diagram|label2|branch_hh|4a4b|nodes}} |2s{4,2,4} |[[W:4,2⁺,4|4,2⁺,4]]||64 | |- align=center !Tetrahedral antiprism |{{Coxeter–Dynkin diagram|node_h|2x|node_h|4|node|3|node}} |s{2,4,3} |[2⁺,4,3]||48 | |- align=center !Alternated square prism prism |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|4|node}} |sr{2,2,4} |[(2,2)⁺,4]||16 | |- align=center !Snub 4-[[W:Orthotope|orthotope]] |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h|2x|node_h}} = {{Coxeter–Dynkin diagram|node_h|2x|node_h|split1-22|nodes_hh}} |s{2^1,1,1} |[2,2,2]⁺ = [2^1,1,1]⁺||8 |{{Coxeter–Dynkin diagram|node_h|2x|node_h|2x|node_h}} |- align=center !rowspan=6|4-[[W:Rhombic fusil|fusil]] |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node|3|node}} |{3,3,4} |[3,3,4]||384 |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1|4|node}} |{4}+{4} or 2{4} |<nowiki>[[W:4,2,4|4,2,4]]</nowiki> = [8,2⁺,8]||128 |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node|2x|node_f1}} |{3,4}+{ } |[4,3,2]||96 |{{Coxeter–Dynkin diagram|node_f1|4|node|3|node}}<br />{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1|2x|node_f1}} |{4}+2{ } |[4,2,2]||32 |{{Coxeter–Dynkin diagram|node_f1|4|node|2x|node_f1}}<br />{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1}} |- align=center |{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1|2x|node_f1}} |{ }+{ }+{ }+{ } or 4{ } |[2,2,2]||16 |{{Coxeter–Dynkin diagram|node_f1|2x|node_f1|2x|node_f1}} |} == Related complex polygons == The [[W:Möbius–Kantor polygon|Möbius–Kantor polygon]] is a [[W:Regular complex polytope|regular complex polygon]] ₃{3}₃, {{Coxeter–Dynkin diagram|3node_1|3|3node}}, in <math>\mathbb{C}^2</math> shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.{{Sfn|Coxeter|1991|pp=30,47}}{{Sfn|Coxeter|Shephard|1992}} The regular complex polygon, ₂{4}₄, {{Coxeter–Dynkin diagram|node_1|4|4node}}, in <math>\mathbb{C}^2</math> has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is ₄[4]₂, order 32.{{Sfn|Coxeter|1991|p=108}} {| class=wikitable width=320 |+ [[W:Orthographic projection|Orthographic projection]]s of ₂{4}₄ polygon |- valign=top |[[File:Complex polygon 2-4-4.png|160px]]<br />In B₄ [[W:Coxeter plane|Coxeter plane]], ₂{4}₄ has 8 vertices and 16 2-edges, shown here with 4 sets of colors. |[[File:Complex polygon 2-4-4 bipartite graph.png|160px]]<br />The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a [[W:Complete bipartite graph|complete bipartite graph]], K_4,4.{{Sfn|Coxeter|1991|p=114}} |} == Related uniform polytopes and honeycombs == The regular 16-cell and [[W:Tesseract|tesseract]] are the regular members of a set of 15 [[W:B4 polytope|uniform 4-polytopes with the same B₄ symmetry]]. The 16-cell is also one of the [[W:D4 polytope|uniform polytopes of D₄ symmetry]]. The 16-cell is also related to the [[W:Cubic honeycomb|cubic honeycomb]], [[W:Order-4 dodecahedral honeycomb|order-4 dodecahedral honeycomb]], and [[W:Order-4 hexagonal tiling honeycomb|order-4 hexagonal tiling honeycomb]] which all have [[W:Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures|octahedral vertex figures]]. It belongs to the sequence of [[W:Order-6 tetrahedral honeycomb#Related polytopes and honeycombs|{3,3,p} 4-polytopes]] which have tetrahedral cells. The sequence includes three [[W:Regular 4-polytope|regular 4-polytope]]s of Euclidean 4-space, the [[5-cell]] {3,3,3}, 16-cell {3,3,4}, and [[600-cell]] {3,3,5}), and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. It is first in a sequence of [[W:Tetrahedral-octahedral honeycomb#Quasuiregular honeycombs|quasiregular polytopes and honeycombs]] h{4,p,q}, and a [[W:Order-4 hexagonal tiling honeycomb#Quasiregular honeycombs|half symmetry sequence]], for regular forms {p,3,4}. == See also == *[[24-cell]] *[[W:4-polytope|4-polytope]] *[[W:D4 polytope|D4 polytope]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * [[W:Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * [[W:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} ** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley] *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] ** {{Cite journal|author-link=W:Harold Scott MacDonald Coxeter|last1=Coxeter|first1=H.S.M.|last2=Shephard|first2=G.C.|title=Portraits of a family of complex polytopes|journal=Leonardo|volume=25|issue=3/4|year=1992|pages=239–244|doi=10.2307/1575843|jstor=1575843|s2cid=124245340}} ** {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }} * [[W:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1_n1) * [[W:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * {{Cite arXiv|eprint=1603.07269|last1=Kim|first1=Heuna|last2=Rote|first2=Günter|title=Congruence Testing of Point Sets in 4 Dimensions|year=2016|class=cs.CG}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} {{Refend}} == External links == * [http://www.polytope.de/c16.html Der 16-Zeller (16-cell)] Marco Möller's Regular polytopes in R⁴ (German) *[http://eusebeia.dyndns.org/4d/16-cell.html Description and diagrams of 16-cell projections] [[Category:Geometry]] [[Category:Polyscheme]] 20iw98mygzirek2huoitfls8mp1wd03 0 305364 2691697 2634795 2024-12-12T22:19:51Z Dc.samizdat 2856930 Categories 2691697 wikitext text/x-wiki {{Short description|Four-dimensional analogue of the tetrahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=5-cell<BR>(4-simplex)| Image_File=Schlegel wireframe 5-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<BR>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Family=[[W:Simplex|Simplex]]| Last= | Index=1| Next=[[W:Rectified 5-cell|2]]| Schläfli={3,3,3}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}| Cell_List=5 [[W:Tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]| Edge_Count= 10| Vertex_Count= 5| Petrie_Polygon=[[W:Pentagon|pentagon]]| Coxeter_Group= A₄, [3,3,3]| Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[W:Tetrahedron|tetrahedron]])| Dual=[[W:Self-dual polytope|Self-dual]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[Image:5-cell.gif|thumb|right|A 3D projection of a 5-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]]] [[File:5-cell net.png|thumb|right|[[W:Net (polyhedron)|Net of five tetrahedra (one hidden)]]]] In [[W:Geometry|geometry]], the '''5-cell''' is the convex [[W:4-polytope|4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,3}. It is a 5-vertex [[W:Four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells.{{Efn|name=elements}} It is also known as a '''C₅''', '''pentachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> '''pentatope''', '''pentahedroid''',<ref>Matila Ghyka, ''The geometry of Art and Life'' (1977), p.68</ref> or '''tetrahedral pyramid'''. It is the '''4-[[W:Simplex|simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{Sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[W:Tetrahedron|tetrahedron]] in three dimensions and the [[W:Triangle|triangle]] in two dimensions. The 5-cell is a [[W:Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides. The '''regular 5-cell''' is bounded by five [[W:Regular tetrahedron|regular tetrahedra]], and is one of the six [[W:Regular convex 4-polytope|regular convex 4-polytope]]s (the four-dimensional analogues of the [[W:Platonic solids|Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions. == Alternative names == * Pentachoron (5-point 4-polytope) * Hypertetrahedron (4-dimensional analogue of the [[W:Tetrahedron|tetrahedron]]) * 4-simplex (4-dimensional [[W:Simplex|simplex]]) * Tetrahedral pyramid (4-dimensional [[W:Hyperpyramid|hyperpyramid]] with a tetrahedral base) * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron)<ref>[http://www.polytope.net/hedrondude/regulars.htm Category 1: Regular Polychora]</ref> ==Geometry== The 5-cell is the 4-dimensional [[W:Simplex|simplex]], the simplest possible [[W:4-polytope|4-polytope]]. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}} {{Regular convex 4-polytopes|wiki=W:}} A 5-cell is formed by any five points which are not all in the same [[W:Hyperplane|hyperplane]] (as a [[W:Tetrahedron|tetrahedron]] is formed by any four points which are not all in the same plane, and a [[W:Triangle|triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[W:Polytope compound|compound]] of 120 regular 5-cells.{{Efn|The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices.}} === Structure === When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. The 5-cell is [[W:Self-dual polytope|self-dual]] (as are all [[W:Simplex|simplexes]]), and its [[W:Vertex figure]|vertex figure]] is the [[W:Tetrahedron|tetrahedron]].{{Efn|The [[W:Schlegel diagram|Schlegel diagram]] of the 5-cell (at the top of this article) illustrates its tetrahedral [[W:Vertex figure|vertex figure]]. Six of the 5-cell's 10 edges are the bounding edges of the Schlegel regular tetrahedron. The other four edges converge at the fifth vertex, at the center of volume of the tetrahedron. Consider any circular geodesic (shortest) path along edges.{{Efn|name=non-planar geodesic circle along edges}} There are four ways to arrive at a vertex (such as that fifth "central" vertex) traveling along an edge. The 5-cell has exactly two distinct pentagonal geodesic circles in it, and the four arrival directions at a vertex correspond to arriving on one of two circuits, traveling in one of two rotational directions on a circuit. These two geodesic skew pentagons are the 5-cell's two distinct [[W:Petrie polygon|Petrie polygon]]s. In the [[#Boerdijk–Coxeter helix|orthogonal projection graph]] one appears as the pentagon perimeter (vertex sequence 1 2 3 4 5), and one is the inscribed pentagram (vertex sequence 1 3 5 2 4), but in fact they are identical regular ''skew'' pentagons, each of which skews through all 4 dimensions. Each is a different sequence of 5 of the 10 edges, and there are only two such distinct sequences.|name=vertex figure}} Its maximal intersection with 3-dimensional space is the [[W:Triangular prism|triangular prism]]. Its [[W:Dihedral angle|dihedral angle]] is cos⁻¹({{sfrac|1|4}}), or approximately 75.52°. The convex hull of two 5-cells in dual configuration is the [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[W:Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]]. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref> [[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]] {| class=wikitable |- !Element||''k''-face||f_''k'' !f₀ !f₁ !f₂ !f₃ !''k''-figs |- align=right |align=left bgcolor=#ffffe0 |<!--. . . .-->{{Coxeter–Dynkin diagram|node_x|2|node_x|2|node_x|2|node_x}}||( ) |rowspan=1|f₀ |bgcolor=#e0ffe0|5 | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |[[W:Tetrahedron|{3,3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x . . .-->{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node_x|2|node_x}}||{ } |rowspan=1|f₁ | bgcolor=#e0e0e0|2 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|3 | bgcolor=#ffffff|3 |[[W:Triangle|{3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x3o . .-->{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node_x}}||[[W:Triangle|{3}]] |rowspan=1|f₂ | bgcolor=#ffffff|3 | bgcolor=#e0e0e0|3 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|2 |{ } |- align=right |align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}}||[[W:Tetrahedron|{3,3}]] |rowspan=1|f₃ | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |bgcolor=#e0ffe0|5 |( ) |} All these elements of the 5-cell are enumerated in [[W:Branko Grünbaum|Branko Grünbaum]]'s [[W:Venn diagram|Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane. ===Coordinates=== The simplest set of [[W:Cartesian coordinates|Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[W:Golden ratio|golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[W:Circumcenter|circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates: :<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi} \right)/(\tfrac{1}{\phi}-2)</math> The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[W:Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space: :<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0, 0, 0,\frac{4}\sqrt{5} \right)</math> Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates: :<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( 0, 0, 0, 1 \right)</math> Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are: :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math> :<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)</math> Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives: :<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math> :<math>\left(-1, 0, 0, 0 \right)</math> The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[W:Hyperplane|hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[W:Facet (geometry)|facet]] of, respectively, the [[W:5-orthoplex|5-orthoplex]] or the [[W:Rectified penteract|rectified penteract]]. === Geodesics and rotations === [[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[W:SO(4)#Double rotations|double rotation]].{{Efn|The [[W:Rotations in 4-dimensional Euclidean space|general rotation in 4-space]] is a [[W:SO(4)#Double rotations|double rotation]], by a distinct angle in each of two completely orthogonal rotation planes. There are two special cases of the double rotation, the [[W:SO(4)#Simple rotations|simple rotation]] (with one 0° rotation angle) and the [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] (with two equal rotation angles).}}]]The 5-cell has only [[W:Digon|digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.{{Efn|In a polytope with a tetrahedral vertex figure,{{Efn|name=vertex figure}} a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path ''[[#Boerdijk–Coxeter helix|Clifford polygon]]'' is the skew chord set of a true geodesic great circle, circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an ''isocline''.{{Efn|name=4-simplex isoclines are edges}}|name=non-planar geodesic circle along edges}} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.{{Efn|Each edge intersects 6 others (3 at each end) and is disjoint from the other 3, to which it is orthogonal as the edge of a tetrahedron to its opposite edge.}} The characteristic [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices. There are only two ways to make a [[W:Hamiltonian circuit|circuit]] of the 5-cell through all 5 vertices along 5 edges,{{Efn|name=vertex figure}} so there are two discrete [[W:Hopf fibration|Hopf fibration]]s of the great digons of the 5-cell. Each of the two fibrations corresponds to a left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only two distinct period 5 ''[[#Boerdijk–Coxeter helix|isoclines]]'' (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations.{{Efn|name=4-simplex isoclines are edges}} Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[W:Clifford torus|Clifford torus]] is depicted in its rectangular (wrapping) form. <gallery caption="[[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]"> File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation </gallery> === Boerdijk–Coxeter helix === A 5-cell can be constructed as a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}} The 10 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[W:Pentagon#Regular pentagons|regular pentagon]] which is the [[W:Petrie polygon|Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[W:Pentagram|pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are the chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].{{Efn|Each isocline chord ([[#Boerdijk–Coxeter helix|blue pentagram edge]]) runs from one of the 5 vertices, through the interior volume of one of the 5 tetrahedral cells, through the cell's triangular face opposite the vertex, and then straight on through the volume of the neighboring cell that shares the face, to its vertex opposite the face. The isocline chord is a straight line between the two vertices through the volume of the two cells. As you can see in [[#Boerdijk–Coxeter helix|the illustration]], the blue isocline chord does not pass through the exact center of the shared face, but rather through a point closer to one face vertex. There are in fact two different isocline pentagrams in the 5-cell, one of which appears as the blue pentagram in the illustration. Each of these two Clifford pentagrams is a different circular sequence of 5 of the 5-cell's 10 edges.{{Efn|name=vertex figure}} All 10 edges are present in each of the 5 tetrahedral cells: each cell is bounded by 6 of the 10 edges, and has the other 4 of the 10 edges running through its volume as isocline chords, from its 4 vertices and through their 4 opposite faces.{{Efn|The 5-cell (4-simplex) is unique among regular 4-polytopes in that its isocline chords{{Efn|name=Clifford polygon}} are its own edges. In the other regular 4-polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5-cell's isocline Clifford polygon (a skew pentagram) and its zig-zag Petrie polygon (a skew pentagon) are exactly the same object; in the other regular 4-polytopes they are quite different.|name=4-simplex isoclines are edges}}|name=Clifford polygon}} :[[File:5-cell 5-ring net.png|480px]] ===Projections=== [[Image:Stereographic polytope 5cell.png|240px|thumb|[[W:Stereographic projection|Stereographic projection]] wireframe (edge projected onto a [[W:3-sphere|3-sphere]])]] The A₄ Coxeter plane projects the 5-cell into a regular [[W:Pentagon|pentagon]] and [[W:Pentagram|pentagram]]. The A₃ Coxeter plane projection of the 5-cell is that of a [[W:Square pyramid|square pyramid]]. The A₂ Coxeter plane projection of the regular 5-cell is that of a [[W:Triangular bipyramid|triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered. {{4-simplex Coxeter plane graphs|t0|150}} {|class="wikitable" width=640 !colspan=2|Projections to 3 dimensions |- valign=top align=center |[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[W:Tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. |[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[W:Triangular dipyramid|triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. |- valign=top align=center |[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection. |[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here. |} == Irregular 5-cells == In the case of [[W:Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[W:Fundamental domain|fundamental domain]]s of the different [[W:Coxeter group|symmetry groups]] which give rise to the various 4-polytopes. ===Orthoschemes=== A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[W:Triangle#By_internal_angles|right triangles]].{{Efn|A 5-cell's 5 vertices form 5 tetrahedral [[W:Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.|name=elements}} An [[W:Schläfli orthoscheme|orthoscheme]] is an irregular [[W:Simplex|simplex]] that is the [[W:Convex hull|convex hull]] of a [[W:Tree (graph theory)|tree]] in which all edges are mutually perpendicular.{{Efn|A right triangle is a 2-dimensional orthoscheme; orthoschemes are the generalization of right triangles to ''n'' dimensions. A 3-dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).}} In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[W:Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle). Orthoschemes are the [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[W:Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[W:4-cube|4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[W:Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[W:Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]]. [[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[W:Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[W:Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).{{Efn|The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.}} The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[W:Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[W:Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge. The 4-cube {{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} can be [[W:Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{Coxeter–Dynkin diagram|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.{{Efn|The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.}} More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[W:Genetic code|genetic code]]s of polytopes: like a [[W:Swiss Army knife|Swiss Army knife]], they contain one of everything needed to construct the polytope by replication. Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{Coxeter–Dynkin diagram|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.{{Efn|The 120 congruent{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations}} 4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite [[W:Chirality|chirality]] (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had right triangles instead of squares).}} {| class="wikitable floatright" !colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small> |align=center|<small>104°30′40″</small> |align=center|<small><math>\pi - 2\text{𝜂}</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>\pi - 2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{30}} \approx 0.183</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{20}} \approx 0.387</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{20}} \approx 0.224</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{60}} \approx 0.129</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>\sqrt{1} = 1.0</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{16}} = 0.25</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.{{Efn|If the regular 5-cell has edge length <small><math>2</math></small> and radius <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>, <small><math>\sqrt{\tfrac{1}{4}} {{=}} 0.5</math></small>, <small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> (the exterior right triangle face, the ''characteristic triangle''), plus <small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small>, <small><math>\sqrt{\tfrac{1}{8}} \approx 0.354</math></small>, <small><math>\sqrt{\tfrac{1}{24}} \approx 0.204</math></small> (the other three edges of the exterior 3-orthoscheme facet the ''characteristic tetrahedron''), plus <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, <small><math>\sqrt{\tfrac{3}{5}} \approx 0.775</math></small>, <small><math>\sqrt{\tfrac{4}{15}} \approx 0.516</math></small>, <small><math>\sqrt{\tfrac{1}{10}} = 0.316</math></small> (edges that are the characteristic radii of the regular 5-cell).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{24}}</math></small>, <small><math>\sqrt{\tfrac{1}{10}}</math></small>.}} ===Isometries=== There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[W:Vertex figure|vertex figure]]s: {| class=wikitable width=600 !Symmetry ![3,3,3]<BR>Order 120 ![3,3,1]<BR>Order 24 ![3,2,1]<BR>Order 12 ![3,1,1]<BR>Order 6 !~[5,2]⁺<BR>Order 10 |- align=center !Name | Regular 5-cell | Tetrahedral [[W:Polyhedral pyramid|pyramid]] | | Triangular pyramidal pyramid | |- align=center ![[W:Schläfli symbol|Schläfli]] | {3,3,3} | {3,3}∨(&nbsp;) | {3}∨{&nbsp;} | {3}∨(&nbsp;)∨(&nbsp;) | |- align=center valign=top !valign=center|Example<BR>Vertex<BR>figure |[[File:5-simplex verf.png|120px]]<BR>[[W:5-simplex|5-simplex]] |[[File:Truncated 5-simplex verf.png|120px]]<BR>[[W:Truncated 5-simplex|Truncated 5-simplex]] |[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[W:Bitruncated 5-simplex|Bitruncated 5-simplex]] |[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[W:Cantitruncated 5-simplex|Cantitruncated 5-simplex]] |[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[W:Omnitruncated 4-simplex honeycomb|Omnitruncated 4-simplex honeycomb]] |} The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[W:Polyhedral pyramid|polyhedral pyramid]], constructed as a regular [[W:Tetrahedron|tetrahedron]] base in a 3-space [[W:Hyperplane|hyperplane]], and an [[W:Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[W:Triangular pyramid|triangular pyramid]] cells. Many [[W:Uniform 5-polytope|uniform 5-polytope]]s have '''tetrahedral pyramid''' [[W:Vertex figure|vertex figure]]s with [[W:Schläfli symbol|Schläfli symbol]]s ( )∨{3,3}. {| class=wikitable |+ Symmetry [3,3,1], order 24 |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:5-cell prism verf.png|100px]] |[[File:Tesseractic prism verf.png|100px]] |[[File:120-cell prism verf.png|100px]] |[[File:Truncated 5-simplex verf.png|100px]] |[[File:Truncated 5-cube verf.png|100px]] |[[File:Truncated 24-cell honeycomb verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:5-cell prism|{ }×{3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|3|node|3|node|3|node}} ![[W:Tesseractic prism|{ }×{4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|4|node|3|node|3|node}} ![[W:120-cell prism|{ }×{5,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|5|node|3|node|3|node}} ![[W:Truncated 5-simplex|t{3,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node|3|node|3|node}} ![[W:Truncated 5-cube|t{4,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node|3|node|3|node}} ![[W:Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|4|node|3|node|3|node}} |} Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[W:Uniform polytope|uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram. {| class=wikitable !Symmetry !colspan=2|[3,2,1], order 12 !colspan=2|[3,1,1], order 6 ![2⁺,4,1], order 8 ![2,1,1], order 4 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Bitruncated 5-simplex verf.png|100px]] |[[File:Bitruncated penteract verf.png|100px]] |[[File:Canitruncated 5-simplex verf.png|100px]] |[[File:Canitruncated 5-cube verf.png|100px]] |[[File:Bicanitruncated 5-simplex verf.png|100px]] |[[File:Bicanitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Bitruncated 5-simplex|t₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node|3|node}} ![[W:Bitruncated 5-cube|t₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-simplex|t₀₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-cube|t₀₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node|3|node}} ![[W:Bicantitruncated 5-simplex|t₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Bicantitruncated 5-cube|t₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node_1|3|node}} |} {| class=wikitable !Symmetry !colspan=3|[2,1,1], order 2 ![2⁺,1,1], order 2 ![ ]⁺, order 1 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Runcicantitruncated 5-simplex verf.png|100px]] |[[File:Runcicantitruncated 5-cube verf.png|100px]] |[[File:Runcicantitruncated 5-orthoplex verf.png|100px]] |[[File:Omnitruncated 5-simplex verf.png|100px]] |[[File:Omnitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Runcicantitruncated 5-simplex|t₀₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-cube|t₀₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-orthoplex|t₀₁₂₃β₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|4|node}} ![[W:Omnitruncated 5-simplex|t₀₁₂₃₄α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} ![[W:Omnitruncated 5-cube|t₀₁₂₃₄γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node_1}} |} == Compound == The compound of two 5-cells in dual configurations can be seen in this A5 [[W:Coxeter plane|Coxeter plane]] projection, with a red and blue 5-cell vertices and edges. This compound has <nowiki>[[W:3,3,3|3,3,3]]</nowiki> symmetry, order 240. The intersection of these two 5-cells is a uniform [[W:Bitruncated 5-cell|bitruncated 5-cell]]. {{Coxeter–Dynkin diagram|branch_11|3ab|nodes}} = {{Coxeter–Dynkin diagram|branch|3ab|nodes_10l}} ∩ {{Coxeter–Dynkin diagram|branch|3ab|nodes_01l}}. :[[File:Compound_dual_5-cells_A5_coxeter_plane.png|240px]] This compound can be seen as the 4D analogue of the 2D [[W:Hexagram|hexagram]] &lbrace;{{sfrac|6|2}}&rbrace; and the 3D [[W:Compound of two tetrahedra|compound of two tetrahedra]]. == Related polytopes and honeycombs == The pentachoron (5-cell) is the simplest of 9 [[W:Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[W:Coxeter group|Coxeter group]]. {{Pentachoron family small}} {{1 k2 polytopes}} {{2 k1 polytopes}} It is in the {p,3,3} sequence of [[W:Regular polychora|regular polychora]] with a [[W:Tetrahedron|tetrahedral]] [[W:Vertex figure|vertex figure]]: the [[W:Tesseract|tesseract]] {4,3,3} and [[120-cell]] {5,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space.{{Efn|name=vertex figure}} {{Tetrahedral vertex figure tessellations small}} It is one of three {3,3,p} [[W:regular 4-polytope]]s with tetrahedral cells, along with the [[16-cell]] {3,3,4} and [[600-cell]] {3,3,5}. The [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space also has tetrahedral cells. {{Tetrahedral cell tessellations}} It is self-dual like the [[24-cell]] {3,4,3}, having a [[W:Palindromic|palindromic]] {3,p,3} [[W:Schläfli symbol|Schläfli symbol]]. {{Symmetric_tessellations}} {{Symmetric2_tessellations}} == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == * [[W:Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * [[W:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} *** p.&nbsp;120, §7.2. see illustration Fig 7.2<small>A</small> *** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * [[W:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1_n1) * [[W:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} ==External links== * [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R⁴ (German) * [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora] * [https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets] * [http://hi.gher.space/wiki/Pyrochoron pyrochoron] [[Category:Geometry]] [[Category:Polyscheme]] kc3lg5jqbr65n280cj7eqcvcgl2k6qg 2691758 2691697 2024-12-13T08:01:41Z Dc.samizdat 2856930 /* Structure */ 2691758 wikitext text/x-wiki {{Short description|Four-dimensional analogue of the tetrahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=5-cell<BR>(4-simplex)| Image_File=Schlegel wireframe 5-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<BR>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Family=[[W:Simplex|Simplex]]| Last= | Index=1| Next=[[W:Rectified 5-cell|2]]| Schläfli={3,3,3}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}| Cell_List=5 [[W:Tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]| Edge_Count= 10| Vertex_Count= 5| Petrie_Polygon=[[W:Pentagon|pentagon]]| Coxeter_Group= A₄, [3,3,3]| Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[W:Tetrahedron|tetrahedron]])| Dual=[[W:Self-dual polytope|Self-dual]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[Image:5-cell.gif|thumb|right|A 3D projection of a 5-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]]] [[File:5-cell net.png|thumb|right|[[W:Net (polyhedron)|Net of five tetrahedra (one hidden)]]]] In [[W:Geometry|geometry]], the '''5-cell''' is the convex [[W:4-polytope|4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,3}. It is a 5-vertex [[W:Four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells.{{Efn|name=elements}} It is also known as a '''C₅''', '''pentachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> '''pentatope''', '''pentahedroid''',<ref>Matila Ghyka, ''The geometry of Art and Life'' (1977), p.68</ref> or '''tetrahedral pyramid'''. It is the '''4-[[W:Simplex|simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{Sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[W:Tetrahedron|tetrahedron]] in three dimensions and the [[W:Triangle|triangle]] in two dimensions. The 5-cell is a [[W:Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides. The '''regular 5-cell''' is bounded by five [[W:Regular tetrahedron|regular tetrahedra]], and is one of the six [[W:Regular convex 4-polytope|regular convex 4-polytope]]s (the four-dimensional analogues of the [[W:Platonic solids|Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions. == Alternative names == * Pentachoron (5-point 4-polytope) * Hypertetrahedron (4-dimensional analogue of the [[W:Tetrahedron|tetrahedron]]) * 4-simplex (4-dimensional [[W:Simplex|simplex]]) * Tetrahedral pyramid (4-dimensional [[W:Hyperpyramid|hyperpyramid]] with a tetrahedral base) * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron)<ref>[http://www.polytope.net/hedrondude/regulars.htm Category 1: Regular Polychora]</ref> ==Geometry== The 5-cell is the 4-dimensional [[W:Simplex|simplex]], the simplest possible [[W:4-polytope|4-polytope]]. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}} {{Regular convex 4-polytopes|wiki=W:}} A 5-cell is formed by any five points which are not all in the same [[W:Hyperplane|hyperplane]] (as a [[W:Tetrahedron|tetrahedron]] is formed by any four points which are not all in the same plane, and a [[W:Triangle|triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[W:Polytope compound|compound]] of 120 regular 5-cells.{{Efn|The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices.}} === Structure === When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. The 5-cell is [[W:Self-dual polytope|self-dual]] (as are all [[W:Simplex|simplexes]]), and its [[W:Vertex figure|vertex figure]] is the [[W:Tetrahedron|tetrahedron]].{{Efn|The [[W:Schlegel diagram|Schlegel diagram]] of the 5-cell (at the top of this article) illustrates its tetrahedral [[W:Vertex figure|vertex figure]]. Six of the 5-cell's 10 edges are the bounding edges of the Schlegel regular tetrahedron. The other four edges converge at the fifth vertex, at the center of volume of the tetrahedron. Consider any circular geodesic (shortest) path along edges.{{Efn|name=non-planar geodesic circle along edges}} There are four ways to arrive at a vertex (such as that fifth "central" vertex) traveling along an edge. The 5-cell has exactly two distinct pentagonal geodesic circles in it, and the four arrival directions at a vertex correspond to arriving on one of two circuits, traveling in one of two rotational directions on a circuit. These two geodesic skew pentagons are the 5-cell's two distinct [[W:Petrie polygon|Petrie polygon]]s. In the [[#Boerdijk–Coxeter helix|orthogonal projection graph]] one appears as the pentagon perimeter (vertex sequence 1 2 3 4 5), and one is the inscribed pentagram (vertex sequence 1 3 5 2 4), but in fact they are identical regular ''skew'' pentagons, each of which skews through all 4 dimensions. Each is a different sequence of 5 of the 10 edges, and there are only two such distinct sequences.|name=vertex figure}} Its maximal intersection with 3-dimensional space is the [[W:Triangular prism|triangular prism]]. Its [[W:Dihedral angle|dihedral angle]] is cos⁻¹({{sfrac|1|4}}), or approximately 75.52°. The convex hull of two 5-cells in dual configuration is the [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[W:Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]]. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref> [[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]] {| class=wikitable |- !Element||''k''-face||f_''k'' !f₀ !f₁ !f₂ !f₃ !''k''-figs |- align=right |align=left bgcolor=#ffffe0 |<!--. . . .-->{{Coxeter–Dynkin diagram|node_x|2|node_x|2|node_x|2|node_x}}||( ) |rowspan=1|f₀ |bgcolor=#e0ffe0|5 | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |[[W:Tetrahedron|{3,3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x . . .-->{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node_x|2|node_x}}||{ } |rowspan=1|f₁ | bgcolor=#e0e0e0|2 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|3 | bgcolor=#ffffff|3 |[[W:Triangle|{3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x3o . .-->{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node_x}}||[[W:Triangle|{3}]] |rowspan=1|f₂ | bgcolor=#ffffff|3 | bgcolor=#e0e0e0|3 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|2 |{ } |- align=right |align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}}||[[W:Tetrahedron|{3,3}]] |rowspan=1|f₃ | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |bgcolor=#e0ffe0|5 |( ) |} All these elements of the 5-cell are enumerated in [[W:Branko Grünbaum|Branko Grünbaum]]'s [[W:Venn diagram|Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane. ===Coordinates=== The simplest set of [[W:Cartesian coordinates|Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[W:Golden ratio|golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[W:Circumcenter|circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates: :<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi} \right)/(\tfrac{1}{\phi}-2)</math> The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[W:Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space: :<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0, 0, 0,\frac{4}\sqrt{5} \right)</math> Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates: :<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( 0, 0, 0, 1 \right)</math> Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are: :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math> :<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)</math> Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives: :<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math> :<math>\left(-1, 0, 0, 0 \right)</math> The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[W:Hyperplane|hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[W:Facet (geometry)|facet]] of, respectively, the [[W:5-orthoplex|5-orthoplex]] or the [[W:Rectified penteract|rectified penteract]]. === Geodesics and rotations === [[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[W:SO(4)#Double rotations|double rotation]].{{Efn|The [[W:Rotations in 4-dimensional Euclidean space|general rotation in 4-space]] is a [[W:SO(4)#Double rotations|double rotation]], by a distinct angle in each of two completely orthogonal rotation planes. There are two special cases of the double rotation, the [[W:SO(4)#Simple rotations|simple rotation]] (with one 0° rotation angle) and the [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] (with two equal rotation angles).}}]]The 5-cell has only [[W:Digon|digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.{{Efn|In a polytope with a tetrahedral vertex figure,{{Efn|name=vertex figure}} a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path ''[[#Boerdijk–Coxeter helix|Clifford polygon]]'' is the skew chord set of a true geodesic great circle, circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an ''isocline''.{{Efn|name=4-simplex isoclines are edges}}|name=non-planar geodesic circle along edges}} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.{{Efn|Each edge intersects 6 others (3 at each end) and is disjoint from the other 3, to which it is orthogonal as the edge of a tetrahedron to its opposite edge.}} The characteristic [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices. There are only two ways to make a [[W:Hamiltonian circuit|circuit]] of the 5-cell through all 5 vertices along 5 edges,{{Efn|name=vertex figure}} so there are two discrete [[W:Hopf fibration|Hopf fibration]]s of the great digons of the 5-cell. Each of the two fibrations corresponds to a left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only two distinct period 5 ''[[#Boerdijk–Coxeter helix|isoclines]]'' (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations.{{Efn|name=4-simplex isoclines are edges}} Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[W:Clifford torus|Clifford torus]] is depicted in its rectangular (wrapping) form. <gallery caption="[[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]"> File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation </gallery> === Boerdijk–Coxeter helix === A 5-cell can be constructed as a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}} The 10 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[W:Pentagon#Regular pentagons|regular pentagon]] which is the [[W:Petrie polygon|Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[W:Pentagram|pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are the chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].{{Efn|Each isocline chord ([[#Boerdijk–Coxeter helix|blue pentagram edge]]) runs from one of the 5 vertices, through the interior volume of one of the 5 tetrahedral cells, through the cell's triangular face opposite the vertex, and then straight on through the volume of the neighboring cell that shares the face, to its vertex opposite the face. The isocline chord is a straight line between the two vertices through the volume of the two cells. As you can see in [[#Boerdijk–Coxeter helix|the illustration]], the blue isocline chord does not pass through the exact center of the shared face, but rather through a point closer to one face vertex. There are in fact two different isocline pentagrams in the 5-cell, one of which appears as the blue pentagram in the illustration. Each of these two Clifford pentagrams is a different circular sequence of 5 of the 5-cell's 10 edges.{{Efn|name=vertex figure}} All 10 edges are present in each of the 5 tetrahedral cells: each cell is bounded by 6 of the 10 edges, and has the other 4 of the 10 edges running through its volume as isocline chords, from its 4 vertices and through their 4 opposite faces.{{Efn|The 5-cell (4-simplex) is unique among regular 4-polytopes in that its isocline chords{{Efn|name=Clifford polygon}} are its own edges. In the other regular 4-polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5-cell's isocline Clifford polygon (a skew pentagram) and its zig-zag Petrie polygon (a skew pentagon) are exactly the same object; in the other regular 4-polytopes they are quite different.|name=4-simplex isoclines are edges}}|name=Clifford polygon}} :[[File:5-cell 5-ring net.png|480px]] ===Projections=== [[Image:Stereographic polytope 5cell.png|240px|thumb|[[W:Stereographic projection|Stereographic projection]] wireframe (edge projected onto a [[W:3-sphere|3-sphere]])]] The A₄ Coxeter plane projects the 5-cell into a regular [[W:Pentagon|pentagon]] and [[W:Pentagram|pentagram]]. The A₃ Coxeter plane projection of the 5-cell is that of a [[W:Square pyramid|square pyramid]]. The A₂ Coxeter plane projection of the regular 5-cell is that of a [[W:Triangular bipyramid|triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered. {{4-simplex Coxeter plane graphs|t0|150}} {|class="wikitable" width=640 !colspan=2|Projections to 3 dimensions |- valign=top align=center |[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[W:Tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. |[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[W:Triangular dipyramid|triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. |- valign=top align=center |[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection. |[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here. |} == Irregular 5-cells == In the case of [[W:Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[W:Fundamental domain|fundamental domain]]s of the different [[W:Coxeter group|symmetry groups]] which give rise to the various 4-polytopes. ===Orthoschemes=== A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[W:Triangle#By_internal_angles|right triangles]].{{Efn|A 5-cell's 5 vertices form 5 tetrahedral [[W:Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.|name=elements}} An [[W:Schläfli orthoscheme|orthoscheme]] is an irregular [[W:Simplex|simplex]] that is the [[W:Convex hull|convex hull]] of a [[W:Tree (graph theory)|tree]] in which all edges are mutually perpendicular.{{Efn|A right triangle is a 2-dimensional orthoscheme; orthoschemes are the generalization of right triangles to ''n'' dimensions. A 3-dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).}} In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[W:Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle). Orthoschemes are the [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[W:Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[W:4-cube|4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[W:Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[W:Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]]. [[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[W:Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[W:Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).{{Efn|The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.}} The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[W:Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[W:Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge. The 4-cube {{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} can be [[W:Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{Coxeter–Dynkin diagram|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.{{Efn|The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.}} More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[W:Genetic code|genetic code]]s of polytopes: like a [[W:Swiss Army knife|Swiss Army knife]], they contain one of everything needed to construct the polytope by replication. Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{Coxeter–Dynkin diagram|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.{{Efn|The 120 congruent{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations}} 4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite [[W:Chirality|chirality]] (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had right triangles instead of squares).}} {| class="wikitable floatright" !colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small> |align=center|<small>104°30′40″</small> |align=center|<small><math>\pi - 2\text{𝜂}</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>\pi - 2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{30}} \approx 0.183</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{20}} \approx 0.387</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{20}} \approx 0.224</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{60}} \approx 0.129</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>\sqrt{1} = 1.0</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{16}} = 0.25</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.{{Efn|If the regular 5-cell has edge length <small><math>2</math></small> and radius <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>, <small><math>\sqrt{\tfrac{1}{4}} {{=}} 0.5</math></small>, <small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> (the exterior right triangle face, the ''characteristic triangle''), plus <small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small>, <small><math>\sqrt{\tfrac{1}{8}} \approx 0.354</math></small>, <small><math>\sqrt{\tfrac{1}{24}} \approx 0.204</math></small> (the other three edges of the exterior 3-orthoscheme facet the ''characteristic tetrahedron''), plus <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, <small><math>\sqrt{\tfrac{3}{5}} \approx 0.775</math></small>, <small><math>\sqrt{\tfrac{4}{15}} \approx 0.516</math></small>, <small><math>\sqrt{\tfrac{1}{10}} = 0.316</math></small> (edges that are the characteristic radii of the regular 5-cell).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{24}}</math></small>, <small><math>\sqrt{\tfrac{1}{10}}</math></small>.}} ===Isometries=== There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[W:Vertex figure|vertex figure]]s: {| class=wikitable width=600 !Symmetry ![3,3,3]<BR>Order 120 ![3,3,1]<BR>Order 24 ![3,2,1]<BR>Order 12 ![3,1,1]<BR>Order 6 !~[5,2]⁺<BR>Order 10 |- align=center !Name | Regular 5-cell | Tetrahedral [[W:Polyhedral pyramid|pyramid]] | | Triangular pyramidal pyramid | |- align=center ![[W:Schläfli symbol|Schläfli]] | {3,3,3} | {3,3}∨(&nbsp;) | {3}∨{&nbsp;} | {3}∨(&nbsp;)∨(&nbsp;) | |- align=center valign=top !valign=center|Example<BR>Vertex<BR>figure |[[File:5-simplex verf.png|120px]]<BR>[[W:5-simplex|5-simplex]] |[[File:Truncated 5-simplex verf.png|120px]]<BR>[[W:Truncated 5-simplex|Truncated 5-simplex]] |[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[W:Bitruncated 5-simplex|Bitruncated 5-simplex]] |[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[W:Cantitruncated 5-simplex|Cantitruncated 5-simplex]] |[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[W:Omnitruncated 4-simplex honeycomb|Omnitruncated 4-simplex honeycomb]] |} The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[W:Polyhedral pyramid|polyhedral pyramid]], constructed as a regular [[W:Tetrahedron|tetrahedron]] base in a 3-space [[W:Hyperplane|hyperplane]], and an [[W:Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[W:Triangular pyramid|triangular pyramid]] cells. Many [[W:Uniform 5-polytope|uniform 5-polytope]]s have '''tetrahedral pyramid''' [[W:Vertex figure|vertex figure]]s with [[W:Schläfli symbol|Schläfli symbol]]s ( )∨{3,3}. {| class=wikitable |+ Symmetry [3,3,1], order 24 |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:5-cell prism verf.png|100px]] |[[File:Tesseractic prism verf.png|100px]] |[[File:120-cell prism verf.png|100px]] |[[File:Truncated 5-simplex verf.png|100px]] |[[File:Truncated 5-cube verf.png|100px]] |[[File:Truncated 24-cell honeycomb verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:5-cell prism|{ }×{3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|3|node|3|node|3|node}} ![[W:Tesseractic prism|{ }×{4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|4|node|3|node|3|node}} ![[W:120-cell prism|{ }×{5,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|5|node|3|node|3|node}} ![[W:Truncated 5-simplex|t{3,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node|3|node|3|node}} ![[W:Truncated 5-cube|t{4,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node|3|node|3|node}} ![[W:Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|4|node|3|node|3|node}} |} Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[W:Uniform polytope|uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram. {| class=wikitable !Symmetry !colspan=2|[3,2,1], order 12 !colspan=2|[3,1,1], order 6 ![2⁺,4,1], order 8 ![2,1,1], order 4 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Bitruncated 5-simplex verf.png|100px]] |[[File:Bitruncated penteract verf.png|100px]] |[[File:Canitruncated 5-simplex verf.png|100px]] |[[File:Canitruncated 5-cube verf.png|100px]] |[[File:Bicanitruncated 5-simplex verf.png|100px]] |[[File:Bicanitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Bitruncated 5-simplex|t₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node|3|node}} ![[W:Bitruncated 5-cube|t₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-simplex|t₀₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-cube|t₀₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node|3|node}} ![[W:Bicantitruncated 5-simplex|t₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Bicantitruncated 5-cube|t₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node_1|3|node}} |} {| class=wikitable !Symmetry !colspan=3|[2,1,1], order 2 ![2⁺,1,1], order 2 ![ ]⁺, order 1 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Runcicantitruncated 5-simplex verf.png|100px]] |[[File:Runcicantitruncated 5-cube verf.png|100px]] |[[File:Runcicantitruncated 5-orthoplex verf.png|100px]] |[[File:Omnitruncated 5-simplex verf.png|100px]] |[[File:Omnitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Runcicantitruncated 5-simplex|t₀₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-cube|t₀₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-orthoplex|t₀₁₂₃β₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|4|node}} ![[W:Omnitruncated 5-simplex|t₀₁₂₃₄α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} ![[W:Omnitruncated 5-cube|t₀₁₂₃₄γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node_1}} |} == Compound == The compound of two 5-cells in dual configurations can be seen in this A5 [[W:Coxeter plane|Coxeter plane]] projection, with a red and blue 5-cell vertices and edges. This compound has <nowiki>[[W:3,3,3|3,3,3]]</nowiki> symmetry, order 240. The intersection of these two 5-cells is a uniform [[W:Bitruncated 5-cell|bitruncated 5-cell]]. {{Coxeter–Dynkin diagram|branch_11|3ab|nodes}} = {{Coxeter–Dynkin diagram|branch|3ab|nodes_10l}} ∩ {{Coxeter–Dynkin diagram|branch|3ab|nodes_01l}}. :[[File:Compound_dual_5-cells_A5_coxeter_plane.png|240px]] This compound can be seen as the 4D analogue of the 2D [[W:Hexagram|hexagram]] &lbrace;{{sfrac|6|2}}&rbrace; and the 3D [[W:Compound of two tetrahedra|compound of two tetrahedra]]. == Related polytopes and honeycombs == The pentachoron (5-cell) is the simplest of 9 [[W:Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[W:Coxeter group|Coxeter group]]. {{Pentachoron family small}} {{1 k2 polytopes}} {{2 k1 polytopes}} It is in the {p,3,3} sequence of [[W:Regular polychora|regular polychora]] with a [[W:Tetrahedron|tetrahedral]] [[W:Vertex figure|vertex figure]]: the [[W:Tesseract|tesseract]] {4,3,3} and [[120-cell]] {5,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space.{{Efn|name=vertex figure}} {{Tetrahedral vertex figure tessellations small}} It is one of three {3,3,p} [[W:regular 4-polytope]]s with tetrahedral cells, along with the [[16-cell]] {3,3,4} and [[600-cell]] {3,3,5}. The [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space also has tetrahedral cells. {{Tetrahedral cell tessellations}} It is self-dual like the [[24-cell]] {3,4,3}, having a [[W:Palindromic|palindromic]] {3,p,3} [[W:Schläfli symbol|Schläfli symbol]]. {{Symmetric_tessellations}} {{Symmetric2_tessellations}} == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == * [[W:Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * [[W:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} *** p.&nbsp;120, §7.2. see illustration Fig 7.2<small>A</small> *** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * [[W:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1_n1) * [[W:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} ==External links== * [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R⁴ (German) * [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora] * [https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets] * [http://hi.gher.space/wiki/Pyrochoron pyrochoron] [[Category:Geometry]] [[Category:Polyscheme]] mzozerozwx6qb0cwaau78gtudtusl9f 2691759 2691758 2024-12-13T08:03:48Z Dc.samizdat 2856930 /* Related polytopes and honeycombs */ 2691759 wikitext text/x-wiki {{Short description|Four-dimensional analogue of the tetrahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=5-cell<BR>(4-simplex)| Image_File=Schlegel wireframe 5-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<BR>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Family=[[W:Simplex|Simplex]]| Last= | Index=1| Next=[[W:Rectified 5-cell|2]]| Schläfli={3,3,3}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}| Cell_List=5 [[W:Tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]| Edge_Count= 10| Vertex_Count= 5| Petrie_Polygon=[[W:Pentagon|pentagon]]| Coxeter_Group= A₄, [3,3,3]| Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[W:Tetrahedron|tetrahedron]])| Dual=[[W:Self-dual polytope|Self-dual]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[Image:5-cell.gif|thumb|right|A 3D projection of a 5-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]]] [[File:5-cell net.png|thumb|right|[[W:Net (polyhedron)|Net of five tetrahedra (one hidden)]]]] In [[W:Geometry|geometry]], the '''5-cell''' is the convex [[W:4-polytope|4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,3}. It is a 5-vertex [[W:Four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells.{{Efn|name=elements}} It is also known as a '''C₅''', '''pentachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> '''pentatope''', '''pentahedroid''',<ref>Matila Ghyka, ''The geometry of Art and Life'' (1977), p.68</ref> or '''tetrahedral pyramid'''. It is the '''4-[[W:Simplex|simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{Sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[W:Tetrahedron|tetrahedron]] in three dimensions and the [[W:Triangle|triangle]] in two dimensions. The 5-cell is a [[W:Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides. The '''regular 5-cell''' is bounded by five [[W:Regular tetrahedron|regular tetrahedra]], and is one of the six [[W:Regular convex 4-polytope|regular convex 4-polytope]]s (the four-dimensional analogues of the [[W:Platonic solids|Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions. == Alternative names == * Pentachoron (5-point 4-polytope) * Hypertetrahedron (4-dimensional analogue of the [[W:Tetrahedron|tetrahedron]]) * 4-simplex (4-dimensional [[W:Simplex|simplex]]) * Tetrahedral pyramid (4-dimensional [[W:Hyperpyramid|hyperpyramid]] with a tetrahedral base) * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron)<ref>[http://www.polytope.net/hedrondude/regulars.htm Category 1: Regular Polychora]</ref> ==Geometry== The 5-cell is the 4-dimensional [[W:Simplex|simplex]], the simplest possible [[W:4-polytope|4-polytope]]. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}} {{Regular convex 4-polytopes|wiki=W:}} A 5-cell is formed by any five points which are not all in the same [[W:Hyperplane|hyperplane]] (as a [[W:Tetrahedron|tetrahedron]] is formed by any four points which are not all in the same plane, and a [[W:Triangle|triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[W:Polytope compound|compound]] of 120 regular 5-cells.{{Efn|The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices.}} === Structure === When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. The 5-cell is [[W:Self-dual polytope|self-dual]] (as are all [[W:Simplex|simplexes]]), and its [[W:Vertex figure|vertex figure]] is the [[W:Tetrahedron|tetrahedron]].{{Efn|The [[W:Schlegel diagram|Schlegel diagram]] of the 5-cell (at the top of this article) illustrates its tetrahedral [[W:Vertex figure|vertex figure]]. Six of the 5-cell's 10 edges are the bounding edges of the Schlegel regular tetrahedron. The other four edges converge at the fifth vertex, at the center of volume of the tetrahedron. Consider any circular geodesic (shortest) path along edges.{{Efn|name=non-planar geodesic circle along edges}} There are four ways to arrive at a vertex (such as that fifth "central" vertex) traveling along an edge. The 5-cell has exactly two distinct pentagonal geodesic circles in it, and the four arrival directions at a vertex correspond to arriving on one of two circuits, traveling in one of two rotational directions on a circuit. These two geodesic skew pentagons are the 5-cell's two distinct [[W:Petrie polygon|Petrie polygon]]s. In the [[#Boerdijk–Coxeter helix|orthogonal projection graph]] one appears as the pentagon perimeter (vertex sequence 1 2 3 4 5), and one is the inscribed pentagram (vertex sequence 1 3 5 2 4), but in fact they are identical regular ''skew'' pentagons, each of which skews through all 4 dimensions. Each is a different sequence of 5 of the 10 edges, and there are only two such distinct sequences.|name=vertex figure}} Its maximal intersection with 3-dimensional space is the [[W:Triangular prism|triangular prism]]. Its [[W:Dihedral angle|dihedral angle]] is cos⁻¹({{sfrac|1|4}}), or approximately 75.52°. The convex hull of two 5-cells in dual configuration is the [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[W:Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]]. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref> [[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]] {| class=wikitable |- !Element||''k''-face||f_''k'' !f₀ !f₁ !f₂ !f₃ !''k''-figs |- align=right |align=left bgcolor=#ffffe0 |<!--. . . .-->{{Coxeter–Dynkin diagram|node_x|2|node_x|2|node_x|2|node_x}}||( ) |rowspan=1|f₀ |bgcolor=#e0ffe0|5 | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |[[W:Tetrahedron|{3,3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x . . .-->{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node_x|2|node_x}}||{ } |rowspan=1|f₁ | bgcolor=#e0e0e0|2 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|3 | bgcolor=#ffffff|3 |[[W:Triangle|{3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x3o . .-->{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node_x}}||[[W:Triangle|{3}]] |rowspan=1|f₂ | bgcolor=#ffffff|3 | bgcolor=#e0e0e0|3 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|2 |{ } |- align=right |align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}}||[[W:Tetrahedron|{3,3}]] |rowspan=1|f₃ | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |bgcolor=#e0ffe0|5 |( ) |} All these elements of the 5-cell are enumerated in [[W:Branko Grünbaum|Branko Grünbaum]]'s [[W:Venn diagram|Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane. ===Coordinates=== The simplest set of [[W:Cartesian coordinates|Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[W:Golden ratio|golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[W:Circumcenter|circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates: :<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi} \right)/(\tfrac{1}{\phi}-2)</math> The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[W:Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space: :<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0, 0, 0,\frac{4}\sqrt{5} \right)</math> Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates: :<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( 0, 0, 0, 1 \right)</math> Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are: :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math> :<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)</math> Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives: :<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math> :<math>\left(-1, 0, 0, 0 \right)</math> The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[W:Hyperplane|hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[W:Facet (geometry)|facet]] of, respectively, the [[W:5-orthoplex|5-orthoplex]] or the [[W:Rectified penteract|rectified penteract]]. === Geodesics and rotations === [[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[W:SO(4)#Double rotations|double rotation]].{{Efn|The [[W:Rotations in 4-dimensional Euclidean space|general rotation in 4-space]] is a [[W:SO(4)#Double rotations|double rotation]], by a distinct angle in each of two completely orthogonal rotation planes. There are two special cases of the double rotation, the [[W:SO(4)#Simple rotations|simple rotation]] (with one 0° rotation angle) and the [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] (with two equal rotation angles).}}]]The 5-cell has only [[W:Digon|digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.{{Efn|In a polytope with a tetrahedral vertex figure,{{Efn|name=vertex figure}} a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path ''[[#Boerdijk–Coxeter helix|Clifford polygon]]'' is the skew chord set of a true geodesic great circle, circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an ''isocline''.{{Efn|name=4-simplex isoclines are edges}}|name=non-planar geodesic circle along edges}} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.{{Efn|Each edge intersects 6 others (3 at each end) and is disjoint from the other 3, to which it is orthogonal as the edge of a tetrahedron to its opposite edge.}} The characteristic [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices. There are only two ways to make a [[W:Hamiltonian circuit|circuit]] of the 5-cell through all 5 vertices along 5 edges,{{Efn|name=vertex figure}} so there are two discrete [[W:Hopf fibration|Hopf fibration]]s of the great digons of the 5-cell. Each of the two fibrations corresponds to a left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only two distinct period 5 ''[[#Boerdijk–Coxeter helix|isoclines]]'' (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations.{{Efn|name=4-simplex isoclines are edges}} Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[W:Clifford torus|Clifford torus]] is depicted in its rectangular (wrapping) form. <gallery caption="[[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]"> File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation </gallery> === Boerdijk–Coxeter helix === A 5-cell can be constructed as a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}} The 10 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[W:Pentagon#Regular pentagons|regular pentagon]] which is the [[W:Petrie polygon|Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[W:Pentagram|pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are the chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].{{Efn|Each isocline chord ([[#Boerdijk–Coxeter helix|blue pentagram edge]]) runs from one of the 5 vertices, through the interior volume of one of the 5 tetrahedral cells, through the cell's triangular face opposite the vertex, and then straight on through the volume of the neighboring cell that shares the face, to its vertex opposite the face. The isocline chord is a straight line between the two vertices through the volume of the two cells. As you can see in [[#Boerdijk–Coxeter helix|the illustration]], the blue isocline chord does not pass through the exact center of the shared face, but rather through a point closer to one face vertex. There are in fact two different isocline pentagrams in the 5-cell, one of which appears as the blue pentagram in the illustration. Each of these two Clifford pentagrams is a different circular sequence of 5 of the 5-cell's 10 edges.{{Efn|name=vertex figure}} All 10 edges are present in each of the 5 tetrahedral cells: each cell is bounded by 6 of the 10 edges, and has the other 4 of the 10 edges running through its volume as isocline chords, from its 4 vertices and through their 4 opposite faces.{{Efn|The 5-cell (4-simplex) is unique among regular 4-polytopes in that its isocline chords{{Efn|name=Clifford polygon}} are its own edges. In the other regular 4-polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5-cell's isocline Clifford polygon (a skew pentagram) and its zig-zag Petrie polygon (a skew pentagon) are exactly the same object; in the other regular 4-polytopes they are quite different.|name=4-simplex isoclines are edges}}|name=Clifford polygon}} :[[File:5-cell 5-ring net.png|480px]] ===Projections=== [[Image:Stereographic polytope 5cell.png|240px|thumb|[[W:Stereographic projection|Stereographic projection]] wireframe (edge projected onto a [[W:3-sphere|3-sphere]])]] The A₄ Coxeter plane projects the 5-cell into a regular [[W:Pentagon|pentagon]] and [[W:Pentagram|pentagram]]. The A₃ Coxeter plane projection of the 5-cell is that of a [[W:Square pyramid|square pyramid]]. The A₂ Coxeter plane projection of the regular 5-cell is that of a [[W:Triangular bipyramid|triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered. {{4-simplex Coxeter plane graphs|t0|150}} {|class="wikitable" width=640 !colspan=2|Projections to 3 dimensions |- valign=top align=center |[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[W:Tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. |[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[W:Triangular dipyramid|triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. |- valign=top align=center |[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection. |[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here. |} == Irregular 5-cells == In the case of [[W:Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[W:Fundamental domain|fundamental domain]]s of the different [[W:Coxeter group|symmetry groups]] which give rise to the various 4-polytopes. ===Orthoschemes=== A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[W:Triangle#By_internal_angles|right triangles]].{{Efn|A 5-cell's 5 vertices form 5 tetrahedral [[W:Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.|name=elements}} An [[W:Schläfli orthoscheme|orthoscheme]] is an irregular [[W:Simplex|simplex]] that is the [[W:Convex hull|convex hull]] of a [[W:Tree (graph theory)|tree]] in which all edges are mutually perpendicular.{{Efn|A right triangle is a 2-dimensional orthoscheme; orthoschemes are the generalization of right triangles to ''n'' dimensions. A 3-dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).}} In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[W:Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle). Orthoschemes are the [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[W:Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[W:4-cube|4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[W:Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[W:Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]]. [[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[W:Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[W:Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).{{Efn|The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.}} The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[W:Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[W:Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge. The 4-cube {{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} can be [[W:Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{Coxeter–Dynkin diagram|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.{{Efn|The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.}} More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[W:Genetic code|genetic code]]s of polytopes: like a [[W:Swiss Army knife|Swiss Army knife]], they contain one of everything needed to construct the polytope by replication. Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{Coxeter–Dynkin diagram|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.{{Efn|The 120 congruent{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations}} 4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite [[W:Chirality|chirality]] (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had right triangles instead of squares).}} {| class="wikitable floatright" !colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small> |align=center|<small>104°30′40″</small> |align=center|<small><math>\pi - 2\text{𝜂}</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>\pi - 2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{30}} \approx 0.183</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{20}} \approx 0.387</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{20}} \approx 0.224</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{60}} \approx 0.129</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>\sqrt{1} = 1.0</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{16}} = 0.25</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.{{Efn|If the regular 5-cell has edge length <small><math>2</math></small> and radius <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>, <small><math>\sqrt{\tfrac{1}{4}} {{=}} 0.5</math></small>, <small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> (the exterior right triangle face, the ''characteristic triangle''), plus <small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small>, <small><math>\sqrt{\tfrac{1}{8}} \approx 0.354</math></small>, <small><math>\sqrt{\tfrac{1}{24}} \approx 0.204</math></small> (the other three edges of the exterior 3-orthoscheme facet the ''characteristic tetrahedron''), plus <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, <small><math>\sqrt{\tfrac{3}{5}} \approx 0.775</math></small>, <small><math>\sqrt{\tfrac{4}{15}} \approx 0.516</math></small>, <small><math>\sqrt{\tfrac{1}{10}} = 0.316</math></small> (edges that are the characteristic radii of the regular 5-cell).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{24}}</math></small>, <small><math>\sqrt{\tfrac{1}{10}}</math></small>.}} ===Isometries=== There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[W:Vertex figure|vertex figure]]s: {| class=wikitable width=600 !Symmetry ![3,3,3]<BR>Order 120 ![3,3,1]<BR>Order 24 ![3,2,1]<BR>Order 12 ![3,1,1]<BR>Order 6 !~[5,2]⁺<BR>Order 10 |- align=center !Name | Regular 5-cell | Tetrahedral [[W:Polyhedral pyramid|pyramid]] | | Triangular pyramidal pyramid | |- align=center ![[W:Schläfli symbol|Schläfli]] | {3,3,3} | {3,3}∨(&nbsp;) | {3}∨{&nbsp;} | {3}∨(&nbsp;)∨(&nbsp;) | |- align=center valign=top !valign=center|Example<BR>Vertex<BR>figure |[[File:5-simplex verf.png|120px]]<BR>[[W:5-simplex|5-simplex]] |[[File:Truncated 5-simplex verf.png|120px]]<BR>[[W:Truncated 5-simplex|Truncated 5-simplex]] |[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[W:Bitruncated 5-simplex|Bitruncated 5-simplex]] |[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[W:Cantitruncated 5-simplex|Cantitruncated 5-simplex]] |[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[W:Omnitruncated 4-simplex honeycomb|Omnitruncated 4-simplex honeycomb]] |} The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[W:Polyhedral pyramid|polyhedral pyramid]], constructed as a regular [[W:Tetrahedron|tetrahedron]] base in a 3-space [[W:Hyperplane|hyperplane]], and an [[W:Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[W:Triangular pyramid|triangular pyramid]] cells. Many [[W:Uniform 5-polytope|uniform 5-polytope]]s have '''tetrahedral pyramid''' [[W:Vertex figure|vertex figure]]s with [[W:Schläfli symbol|Schläfli symbol]]s ( )∨{3,3}. {| class=wikitable |+ Symmetry [3,3,1], order 24 |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:5-cell prism verf.png|100px]] |[[File:Tesseractic prism verf.png|100px]] |[[File:120-cell prism verf.png|100px]] |[[File:Truncated 5-simplex verf.png|100px]] |[[File:Truncated 5-cube verf.png|100px]] |[[File:Truncated 24-cell honeycomb verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:5-cell prism|{ }×{3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|3|node|3|node|3|node}} ![[W:Tesseractic prism|{ }×{4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|4|node|3|node|3|node}} ![[W:120-cell prism|{ }×{5,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|5|node|3|node|3|node}} ![[W:Truncated 5-simplex|t{3,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node|3|node|3|node}} ![[W:Truncated 5-cube|t{4,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node|3|node|3|node}} ![[W:Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|4|node|3|node|3|node}} |} Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[W:Uniform polytope|uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram. {| class=wikitable !Symmetry !colspan=2|[3,2,1], order 12 !colspan=2|[3,1,1], order 6 ![2⁺,4,1], order 8 ![2,1,1], order 4 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Bitruncated 5-simplex verf.png|100px]] |[[File:Bitruncated penteract verf.png|100px]] |[[File:Canitruncated 5-simplex verf.png|100px]] |[[File:Canitruncated 5-cube verf.png|100px]] |[[File:Bicanitruncated 5-simplex verf.png|100px]] |[[File:Bicanitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Bitruncated 5-simplex|t₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node|3|node}} ![[W:Bitruncated 5-cube|t₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-simplex|t₀₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-cube|t₀₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node|3|node}} ![[W:Bicantitruncated 5-simplex|t₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Bicantitruncated 5-cube|t₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node_1|3|node}} |} {| class=wikitable !Symmetry !colspan=3|[2,1,1], order 2 ![2⁺,1,1], order 2 ![ ]⁺, order 1 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Runcicantitruncated 5-simplex verf.png|100px]] |[[File:Runcicantitruncated 5-cube verf.png|100px]] |[[File:Runcicantitruncated 5-orthoplex verf.png|100px]] |[[File:Omnitruncated 5-simplex verf.png|100px]] |[[File:Omnitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Runcicantitruncated 5-simplex|t₀₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-cube|t₀₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-orthoplex|t₀₁₂₃β₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|4|node}} ![[W:Omnitruncated 5-simplex|t₀₁₂₃₄α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} ![[W:Omnitruncated 5-cube|t₀₁₂₃₄γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node_1}} |} == Compound == The compound of two 5-cells in dual configurations can be seen in this A5 [[W:Coxeter plane|Coxeter plane]] projection, with a red and blue 5-cell vertices and edges. This compound has <nowiki>[[W:3,3,3|3,3,3]]</nowiki> symmetry, order 240. The intersection of these two 5-cells is a uniform [[W:Bitruncated 5-cell|bitruncated 5-cell]]. {{Coxeter–Dynkin diagram|branch_11|3ab|nodes}} = {{Coxeter–Dynkin diagram|branch|3ab|nodes_10l}} ∩ {{Coxeter–Dynkin diagram|branch|3ab|nodes_01l}}. :[[File:Compound_dual_5-cells_A5_coxeter_plane.png|240px]] This compound can be seen as the 4D analogue of the 2D [[W:Hexagram|hexagram]] &lbrace;{{sfrac|6|2}}&rbrace; and the 3D [[W:Compound of two tetrahedra|compound of two tetrahedra]]. == Related polytopes and honeycombs == The pentachoron (5-cell) is the simplest of 9 [[W:Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[W:Coxeter group|Coxeter group]]. {{Pentachoron family small}} {{1 k2 polytopes}} {{2 k1 polytopes}} It is in the {p,3,3} sequence of [[W:Regular polychora|regular polychora]] with a [[W:Tetrahedron|tetrahedral]] [[W:Vertex figure|vertex figure]]: the [[W:Tesseract|tesseract]] {4,3,3} and [[120-cell]] {5,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space.{{Efn|name=vertex figure}} {{Tetrahedral vertex figure tessellations small}} It is one of three {3,3,p} [[W:Regular 4-polytope|regular 4-polytope]]s with tetrahedral cells, along with the [[16-cell]] {3,3,4} and [[600-cell]] {3,3,5}. The [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space also has tetrahedral cells. {{Tetrahedral cell tessellations}} It is self-dual like the [[24-cell]] {3,4,3}, having a [[W:Palindromic|palindromic]] {3,p,3} [[W:Schläfli symbol|Schläfli symbol]]. {{Symmetric_tessellations}} {{Symmetric2_tessellations}} == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == * [[W:Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * [[W:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} *** p.&nbsp;120, §7.2. see illustration Fig 7.2<small>A</small> *** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * [[W:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1_n1) * [[W:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} ==External links== * [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R⁴ (German) * [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora] * [https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets] * [http://hi.gher.space/wiki/Pyrochoron pyrochoron] [[Category:Geometry]] [[Category:Polyscheme]] 3cyjhcrcnekgqihh6nl0ms5ht9n9hu9 2691760 2691759 2024-12-13T08:04:35Z Dc.samizdat 2856930 /* Compound */ 2691760 wikitext text/x-wiki {{Short description|Four-dimensional analogue of the tetrahedron}} {{Polyscheme|radius=an '''expanded version''' of}} {{Infobox 4-polytope | Name=5-cell<BR>(4-simplex)| Image_File=Schlegel wireframe 5-cell.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<BR>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Family=[[W:Simplex|Simplex]]| Last= | Index=1| Next=[[W:Rectified 5-cell|2]]| Schläfli={3,3,3}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}| Cell_List=5 [[W:Tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]| Edge_Count= 10| Vertex_Count= 5| Petrie_Polygon=[[W:Pentagon|pentagon]]| Coxeter_Group= A₄, [3,3,3]| Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[W:Tetrahedron|tetrahedron]])| Dual=[[W:Self-dual polytope|Self-dual]]| Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[Image:5-cell.gif|thumb|right|A 3D projection of a 5-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]]] [[File:5-cell net.png|thumb|right|[[W:Net (polyhedron)|Net of five tetrahedra (one hidden)]]]] In [[W:Geometry|geometry]], the '''5-cell''' is the convex [[W:4-polytope|4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,3}. It is a 5-vertex [[W:Four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells.{{Efn|name=elements}} It is also known as a '''C₅''', '''pentachoron''',<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> '''pentatope''', '''pentahedroid''',<ref>Matila Ghyka, ''The geometry of Art and Life'' (1977), p.68</ref> or '''tetrahedral pyramid'''. It is the '''4-[[W:Simplex|simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{Sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[W:Tetrahedron|tetrahedron]] in three dimensions and the [[W:Triangle|triangle]] in two dimensions. The 5-cell is a [[W:Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides. The '''regular 5-cell''' is bounded by five [[W:Regular tetrahedron|regular tetrahedra]], and is one of the six [[W:Regular convex 4-polytope|regular convex 4-polytope]]s (the four-dimensional analogues of the [[W:Platonic solids|Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions. == Alternative names == * Pentachoron (5-point 4-polytope) * Hypertetrahedron (4-dimensional analogue of the [[W:Tetrahedron|tetrahedron]]) * 4-simplex (4-dimensional [[W:Simplex|simplex]]) * Tetrahedral pyramid (4-dimensional [[W:Hyperpyramid|hyperpyramid]] with a tetrahedral base) * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron)<ref>[http://www.polytope.net/hedrondude/regulars.htm Category 1: Regular Polychora]</ref> ==Geometry== The 5-cell is the 4-dimensional [[W:Simplex|simplex]], the simplest possible [[W:4-polytope|4-polytope]]. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}} {{Regular convex 4-polytopes|wiki=W:}} A 5-cell is formed by any five points which are not all in the same [[W:Hyperplane|hyperplane]] (as a [[W:Tetrahedron|tetrahedron]] is formed by any four points which are not all in the same plane, and a [[W:Triangle|triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[W:Polytope compound|compound]] of 120 regular 5-cells.{{Efn|The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices.}} === Structure === When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. The 5-cell is [[W:Self-dual polytope|self-dual]] (as are all [[W:Simplex|simplexes]]), and its [[W:Vertex figure|vertex figure]] is the [[W:Tetrahedron|tetrahedron]].{{Efn|The [[W:Schlegel diagram|Schlegel diagram]] of the 5-cell (at the top of this article) illustrates its tetrahedral [[W:Vertex figure|vertex figure]]. Six of the 5-cell's 10 edges are the bounding edges of the Schlegel regular tetrahedron. The other four edges converge at the fifth vertex, at the center of volume of the tetrahedron. Consider any circular geodesic (shortest) path along edges.{{Efn|name=non-planar geodesic circle along edges}} There are four ways to arrive at a vertex (such as that fifth "central" vertex) traveling along an edge. The 5-cell has exactly two distinct pentagonal geodesic circles in it, and the four arrival directions at a vertex correspond to arriving on one of two circuits, traveling in one of two rotational directions on a circuit. These two geodesic skew pentagons are the 5-cell's two distinct [[W:Petrie polygon|Petrie polygon]]s. In the [[#Boerdijk–Coxeter helix|orthogonal projection graph]] one appears as the pentagon perimeter (vertex sequence 1 2 3 4 5), and one is the inscribed pentagram (vertex sequence 1 3 5 2 4), but in fact they are identical regular ''skew'' pentagons, each of which skews through all 4 dimensions. Each is a different sequence of 5 of the 10 edges, and there are only two such distinct sequences.|name=vertex figure}} Its maximal intersection with 3-dimensional space is the [[W:Triangular prism|triangular prism]]. Its [[W:Dihedral angle|dihedral angle]] is cos⁻¹({{sfrac|1|4}}), or approximately 75.52°. The convex hull of two 5-cells in dual configuration is the [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[W:Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]]. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref> [[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]] {| class=wikitable |- !Element||''k''-face||f_''k'' !f₀ !f₁ !f₂ !f₃ !''k''-figs |- align=right |align=left bgcolor=#ffffe0 |<!--. . . .-->{{Coxeter–Dynkin diagram|node_x|2|node_x|2|node_x|2|node_x}}||( ) |rowspan=1|f₀ |bgcolor=#e0ffe0|5 | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |[[W:Tetrahedron|{3,3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x . . .-->{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node_x|2|node_x}}||{ } |rowspan=1|f₁ | bgcolor=#e0e0e0|2 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|3 | bgcolor=#ffffff|3 |[[W:Triangle|{3}]] |- align=right |align=left bgcolor=#ffffe0 |<!--x3o . .-->{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node_x}}||[[W:Triangle|{3}]] |rowspan=1|f₂ | bgcolor=#ffffff|3 | bgcolor=#e0e0e0|3 |bgcolor=#e0ffe0|10 | bgcolor=#e0e0e0|2 |{ } |- align=right |align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}}||[[W:Tetrahedron|{3,3}]] |rowspan=1|f₃ | bgcolor=#e0e0e0|4 | bgcolor=#ffffff|6 | bgcolor=#e0e0e0|4 |bgcolor=#e0ffe0|5 |( ) |} All these elements of the 5-cell are enumerated in [[W:Branko Grünbaum|Branko Grünbaum]]'s [[W:Venn diagram|Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane. ===Coordinates=== The simplest set of [[W:Cartesian coordinates|Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[W:Golden ratio|golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[W:Circumcenter|circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates: :<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math> :<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi} \right)/(\tfrac{1}{\phi}-2)</math> The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[W:Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space: :<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math> :<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math> :<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math> :<math>\left( 0, 0, 0,\frac{4}\sqrt{5} \right)</math> Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates: :<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math> :<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math> :<math>\left( 0, 0, 0, 1 \right)</math> Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are: :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math> :<math>\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math> :<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)</math> Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives: :<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math> :<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math> :<math>\left(-1, 0, 0, 0 \right)</math> The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[W:Hyperplane|hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[W:Facet (geometry)|facet]] of, respectively, the [[W:5-orthoplex|5-orthoplex]] or the [[W:Rectified penteract|rectified penteract]]. === Geodesics and rotations === [[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[W:SO(4)#Double rotations|double rotation]].{{Efn|The [[W:Rotations in 4-dimensional Euclidean space|general rotation in 4-space]] is a [[W:SO(4)#Double rotations|double rotation]], by a distinct angle in each of two completely orthogonal rotation planes. There are two special cases of the double rotation, the [[W:SO(4)#Simple rotations|simple rotation]] (with one 0° rotation angle) and the [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] (with two equal rotation angles).}}]]The 5-cell has only [[W:Digon|digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.{{Efn|In a polytope with a tetrahedral vertex figure,{{Efn|name=vertex figure}} a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path ''[[#Boerdijk–Coxeter helix|Clifford polygon]]'' is the skew chord set of a true geodesic great circle, circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an ''isocline''.{{Efn|name=4-simplex isoclines are edges}}|name=non-planar geodesic circle along edges}} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.{{Efn|Each edge intersects 6 others (3 at each end) and is disjoint from the other 3, to which it is orthogonal as the edge of a tetrahedron to its opposite edge.}} The characteristic [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices. There are only two ways to make a [[W:Hamiltonian circuit|circuit]] of the 5-cell through all 5 vertices along 5 edges,{{Efn|name=vertex figure}} so there are two discrete [[W:Hopf fibration|Hopf fibration]]s of the great digons of the 5-cell. Each of the two fibrations corresponds to a left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only two distinct period 5 ''[[#Boerdijk–Coxeter helix|isoclines]]'' (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations.{{Efn|name=4-simplex isoclines are edges}} Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[W:Clifford torus|Clifford torus]] is depicted in its rectangular (wrapping) form. <gallery caption="[[W:Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]"> File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation </gallery> === Boerdijk–Coxeter helix === A 5-cell can be constructed as a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}} The 10 triangle faces can be seen in a 2D net within a [[W:Triangular tiling|triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[W:Pentagon#Regular pentagons|regular pentagon]] which is the [[W:Petrie polygon|Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[W:Pentagram|pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are the chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].{{Efn|Each isocline chord ([[#Boerdijk–Coxeter helix|blue pentagram edge]]) runs from one of the 5 vertices, through the interior volume of one of the 5 tetrahedral cells, through the cell's triangular face opposite the vertex, and then straight on through the volume of the neighboring cell that shares the face, to its vertex opposite the face. The isocline chord is a straight line between the two vertices through the volume of the two cells. As you can see in [[#Boerdijk–Coxeter helix|the illustration]], the blue isocline chord does not pass through the exact center of the shared face, but rather through a point closer to one face vertex. There are in fact two different isocline pentagrams in the 5-cell, one of which appears as the blue pentagram in the illustration. Each of these two Clifford pentagrams is a different circular sequence of 5 of the 5-cell's 10 edges.{{Efn|name=vertex figure}} All 10 edges are present in each of the 5 tetrahedral cells: each cell is bounded by 6 of the 10 edges, and has the other 4 of the 10 edges running through its volume as isocline chords, from its 4 vertices and through their 4 opposite faces.{{Efn|The 5-cell (4-simplex) is unique among regular 4-polytopes in that its isocline chords{{Efn|name=Clifford polygon}} are its own edges. In the other regular 4-polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5-cell's isocline Clifford polygon (a skew pentagram) and its zig-zag Petrie polygon (a skew pentagon) are exactly the same object; in the other regular 4-polytopes they are quite different.|name=4-simplex isoclines are edges}}|name=Clifford polygon}} :[[File:5-cell 5-ring net.png|480px]] ===Projections=== [[Image:Stereographic polytope 5cell.png|240px|thumb|[[W:Stereographic projection|Stereographic projection]] wireframe (edge projected onto a [[W:3-sphere|3-sphere]])]] The A₄ Coxeter plane projects the 5-cell into a regular [[W:Pentagon|pentagon]] and [[W:Pentagram|pentagram]]. The A₃ Coxeter plane projection of the 5-cell is that of a [[W:Square pyramid|square pyramid]]. The A₂ Coxeter plane projection of the regular 5-cell is that of a [[W:Triangular bipyramid|triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered. {{4-simplex Coxeter plane graphs|t0|150}} {|class="wikitable" width=640 !colspan=2|Projections to 3 dimensions |- valign=top align=center |[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[W:Tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. |[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[W:Triangular dipyramid|triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. |- valign=top align=center |[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection. |[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here. |} == Irregular 5-cells == In the case of [[W:Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[W:Fundamental domain|fundamental domain]]s of the different [[W:Coxeter group|symmetry groups]] which give rise to the various 4-polytopes. ===Orthoschemes=== A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[W:Triangle#By_internal_angles|right triangles]].{{Efn|A 5-cell's 5 vertices form 5 tetrahedral [[W:Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.|name=elements}} An [[W:Schläfli orthoscheme|orthoscheme]] is an irregular [[W:Simplex|simplex]] that is the [[W:Convex hull|convex hull]] of a [[W:Tree (graph theory)|tree]] in which all edges are mutually perpendicular.{{Efn|A right triangle is a 2-dimensional orthoscheme; orthoschemes are the generalization of right triangles to ''n'' dimensions. A 3-dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).}} In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[W:Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle). Orthoschemes are the [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[W:Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[W:4-cube|4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[W:Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[W:Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]]. [[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[W:Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[W:Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).{{Efn|The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.}} The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[W:Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[W:Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge. The 4-cube {{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}} can be [[W:Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{Coxeter–Dynkin diagram|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.{{Efn|The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.}} More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[W:Genetic code|genetic code]]s of polytopes: like a [[W:Swiss Army knife|Swiss Army knife]], they contain one of everything needed to construct the polytope by replication. Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}} There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{Coxeter–Dynkin diagram|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.{{Efn|The 120 congruent{{Sfn|Coxeter|1973|loc=§3.1 Congruent transformations}} 4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite [[W:Chirality|chirality]] (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had right triangles instead of squares).}} {| class="wikitable floatright" !colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small> |align=center|<small>104°30′40″</small> |align=center|<small><math>\pi - 2\text{𝜂}</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>\pi - 2\text{𝟁}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{30}} \approx 0.183</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{20}} \approx 0.387</math></small> |align=center|<small>75°29′20″</small> |align=center|<small><math>2\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{20}} \approx 0.224</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{60}} \approx 0.129</math></small> |align=center|<small>52°15′20″</small> |align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>\sqrt{1} = 1.0</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{16}} = 0.25</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\text{𝜼}</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.{{Efn|If the regular 5-cell has edge length <small><math>2</math></small> and radius <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>, <small><math>\sqrt{\tfrac{1}{4}} {{=}} 0.5</math></small>, <small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> (the exterior right triangle face, the ''characteristic triangle''), plus <small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small>, <small><math>\sqrt{\tfrac{1}{8}} \approx 0.354</math></small>, <small><math>\sqrt{\tfrac{1}{24}} \approx 0.204</math></small> (the other three edges of the exterior 3-orthoscheme facet the ''characteristic tetrahedron''), plus <small><math>2\sqrt{\tfrac{2}{5}} \approx 1.265</math></small>, <small><math>\sqrt{\tfrac{3}{5}} \approx 0.775</math></small>, <small><math>\sqrt{\tfrac{4}{15}} \approx 0.516</math></small>, <small><math>\sqrt{\tfrac{1}{10}} = 0.316</math></small> (edges that are the characteristic radii of the regular 5-cell).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼₄"}} The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{24}}</math></small>, <small><math>\sqrt{\tfrac{1}{10}}</math></small>.}} ===Isometries=== There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[W:Vertex figure|vertex figure]]s: {| class=wikitable width=600 !Symmetry ![3,3,3]<BR>Order 120 ![3,3,1]<BR>Order 24 ![3,2,1]<BR>Order 12 ![3,1,1]<BR>Order 6 !~[5,2]⁺<BR>Order 10 |- align=center !Name | Regular 5-cell | Tetrahedral [[W:Polyhedral pyramid|pyramid]] | | Triangular pyramidal pyramid | |- align=center ![[W:Schläfli symbol|Schläfli]] | {3,3,3} | {3,3}∨(&nbsp;) | {3}∨{&nbsp;} | {3}∨(&nbsp;)∨(&nbsp;) | |- align=center valign=top !valign=center|Example<BR>Vertex<BR>figure |[[File:5-simplex verf.png|120px]]<BR>[[W:5-simplex|5-simplex]] |[[File:Truncated 5-simplex verf.png|120px]]<BR>[[W:Truncated 5-simplex|Truncated 5-simplex]] |[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[W:Bitruncated 5-simplex|Bitruncated 5-simplex]] |[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[W:Cantitruncated 5-simplex|Cantitruncated 5-simplex]] |[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[W:Omnitruncated 4-simplex honeycomb|Omnitruncated 4-simplex honeycomb]] |} The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[W:Polyhedral pyramid|polyhedral pyramid]], constructed as a regular [[W:Tetrahedron|tetrahedron]] base in a 3-space [[W:Hyperplane|hyperplane]], and an [[W:Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[W:Triangular pyramid|triangular pyramid]] cells. Many [[W:Uniform 5-polytope|uniform 5-polytope]]s have '''tetrahedral pyramid''' [[W:Vertex figure|vertex figure]]s with [[W:Schläfli symbol|Schläfli symbol]]s ( )∨{3,3}. {| class=wikitable |+ Symmetry [3,3,1], order 24 |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:5-cell prism verf.png|100px]] |[[File:Tesseractic prism verf.png|100px]] |[[File:120-cell prism verf.png|100px]] |[[File:Truncated 5-simplex verf.png|100px]] |[[File:Truncated 5-cube verf.png|100px]] |[[File:Truncated 24-cell honeycomb verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:5-cell prism|{ }×{3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|3|node|3|node|3|node}} ![[W:Tesseractic prism|{ }×{4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|4|node|3|node|3|node}} ![[W:120-cell prism|{ }×{5,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|2|node_1|5|node|3|node|3|node}} ![[W:Truncated 5-simplex|t{3,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node|3|node|3|node}} ![[W:Truncated 5-cube|t{4,3,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node|3|node|3|node}} ![[W:Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|4|node|3|node|3|node}} |} Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[W:Uniform polytope|uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram. {| class=wikitable !Symmetry !colspan=2|[3,2,1], order 12 !colspan=2|[3,1,1], order 6 ![2⁺,4,1], order 8 ![2,1,1], order 4 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Bitruncated 5-simplex verf.png|100px]] |[[File:Bitruncated penteract verf.png|100px]] |[[File:Canitruncated 5-simplex verf.png|100px]] |[[File:Canitruncated 5-cube verf.png|100px]] |[[File:Bicanitruncated 5-simplex verf.png|100px]] |[[File:Bicanitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Bitruncated 5-simplex|t₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node|3|node}} ![[W:Bitruncated 5-cube|t₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-simplex|t₀₁₂α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node|3|node}} ![[W:Cantitruncated 5-cube|t₀₁₂γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node|3|node}} ![[W:Bicantitruncated 5-simplex|t₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Bicantitruncated 5-cube|t₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node|4|node_1|3|node_1|3|node_1|3|node}} |} {| class=wikitable !Symmetry !colspan=3|[2,1,1], order 2 ![2⁺,1,1], order 2 ![ ]⁺, order 1 |- align=center ![[W:Schläfli symbol|Schläfli]] |colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;) |- ![[W:Schlegel diagram|Schlegel<BR>diagram]] |[[File:Runcicantitruncated 5-simplex verf.png|100px]] |[[File:Runcicantitruncated 5-cube verf.png|100px]] |[[File:Runcicantitruncated 5-orthoplex verf.png|100px]] |[[File:Omnitruncated 5-simplex verf.png|100px]] |[[File:Omnitruncated 5-cube verf.png|100px]] |- !Name<BR>[[W:Coxeter diagram|Coxeter]] ![[W:Runcicantitruncated 5-simplex|t₀₁₂₃α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-cube|t₀₁₂₃γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node}} ![[W:Runcicantitruncated 5-orthoplex|t₀₁₂₃β₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|4|node}} ![[W:Omnitruncated 5-simplex|t₀₁₂₃₄α₅]]<BR>{{Coxeter–Dynkin diagram|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} ![[W:Omnitruncated 5-cube|t₀₁₂₃₄γ₅]]<BR>{{Coxeter–Dynkin diagram|node_1|4|node_1|3|node_1|3|node_1|3|node_1}} |} == Compound == The compound of two 5-cells in dual configurations can be seen in this A5 [[W:Coxeter plane|Coxeter plane]] projection, with a red and blue 5-cell vertices and edges. This compound has [[W:3,3,3|3,3,3]] symmetry, order 240. The intersection of these two 5-cells is a uniform [[W:Bitruncated 5-cell|bitruncated 5-cell]]. {{Coxeter–Dynkin diagram|branch_11|3ab|nodes}} = {{Coxeter–Dynkin diagram|branch|3ab|nodes_10l}} ∩ {{Coxeter–Dynkin diagram|branch|3ab|nodes_01l}}. :[[File:Compound_dual_5-cells_A5_coxeter_plane.png|240px]] This compound can be seen as the 4D analogue of the 2D [[W:Hexagram|hexagram]] &lbrace;{{sfrac|6|2}}&rbrace; and the 3D [[W:Compound of two tetrahedra|compound of two tetrahedra]]. == Related polytopes and honeycombs == The pentachoron (5-cell) is the simplest of 9 [[W:Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[W:Coxeter group|Coxeter group]]. {{Pentachoron family small}} {{1 k2 polytopes}} {{2 k1 polytopes}} It is in the {p,3,3} sequence of [[W:Regular polychora|regular polychora]] with a [[W:Tetrahedron|tetrahedral]] [[W:Vertex figure|vertex figure]]: the [[W:Tesseract|tesseract]] {4,3,3} and [[120-cell]] {5,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space.{{Efn|name=vertex figure}} {{Tetrahedral vertex figure tessellations small}} It is one of three {3,3,p} [[W:Regular 4-polytope|regular 4-polytope]]s with tetrahedral cells, along with the [[16-cell]] {3,3,4} and [[600-cell]] {3,3,5}. The [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space also has tetrahedral cells. {{Tetrahedral cell tessellations}} It is self-dual like the [[24-cell]] {3,4,3}, having a [[W:Palindromic|palindromic]] {3,p,3} [[W:Schläfli symbol|Schläfli symbol]]. {{Symmetric_tessellations}} {{Symmetric2_tessellations}} == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == * [[W:Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * [[W:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} *** p.&nbsp;120, §7.2. see illustration Fig 7.2<small>A</small> *** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * [[W:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1_n1) * [[W:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} ==External links== * [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R⁴ (German) * [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora] * [https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets] * [http://hi.gher.space/wiki/Pyrochoron pyrochoron] [[Category:Geometry]] [[Category:Polyscheme]] 722lnw8ncqaw2bmkwmew4imw4xqfsft 0 306269 2691670 2691085 2024-12-12T17:57:40Z Parodda 2936296 /* Variable, Subscale, and Total Score Information */ Adding subscale information and notes 2691670 wikitext text/x-wiki === Adolescent Brain Cognitive Development (ABCD) Study === [https://abcdstudy.org/study-sites/ The Adolescent Brain Cognitive Development (ABCD) Study'''©'''] is the largest long-term study of brain development and child health in the United States, with 11,880 children between the ages of 9 and 10 having been invited to join the study. The ABCD study is funded by the National Institutes of Health (NIH) and is associated with 21 research sites across the country. Researchers seek to explore how childhood experiences interact to affect cognitive, social, emotional, and physical development during adolescence. The goal of the ABCD study is to provide families, schools, professionals, and policy makers with crucial information needed to promote the health, well-being, and success of children. ''Copyright '''©''' 2024 ABCD Study | "ABCD Study'''©''', Teen Brains. Today's Science. Brighter Future.'''©''', El cerebro adolescente. La ciencia de hoy. Un futuro más brillante.'''©''' and the ABCD Study Logos are registered marks of the U.S. Department of Health & Human Services (HHS). Adolescent Brain Cognitive Development℠ Study, El Estudio del Desarrollo Cognitivo y Cerebral del Adolescente℠, are service marks of the U.S. Department of Health & Human Services (HHS)."'' ==== ABCD Coordinating Center ==== Principal Investigators: Drs. Terry Jernigan and Sandra A. Brown Manager: David Benjamin Email: abcd-cc@ucsd.edu ====== Data Analysis, Informatics & Resource Center (DAIRC) ====== Principal Investigator: Dr. Anders Dale Associate Director, Bioinformatics: Dr. Rongguang Yang Email: roy002@health.ucsd.edu == Materials == This page combines and organizes materials from the following sources: === ABCD Protocol by Wave === The following are ABCD Protocol for both youth and parents by wave. All PDFs were created by the ABCD study and were the first resource used to build this page. * [https://abcdstudy.org/wp-content/uploads/2019/12/flyer_protocol_Baseline_new.pdf ABCD Protocol Summary: Baseline] * [https://abcdstudy.org/wp-content/uploads/2019/12/flyer_protocol-1yrFlup_eg_both.pdf ABCD Protocol Summary: One-year Follow-up] * [https://abcdstudy.org/wp-content/uploads/2021/11/flyer_protocol-2yrFlup_eg_both.pdf ABCD Protocol Summary: Two-year Follow-up] * [https://abcdstudy.org/wp-content/uploads/2020/11/flyer_protocol-3yrFlup_eg.pdf ABCD Protocol Summary: Three-year Follow-up] * [https://abcdstudy.org/wp-content/uploads/2022/05/flyer_protocol-4yrFlup_eg-final.pdf ABCD Protocol Summary: Four-year Follow-up] * [https://abcdstudy.org/wp-content/uploads/2023/07/ABCD_Youth_Protocol_Summary_Five-Year_Follow-Up_Flyer_041823.pdf ABCD Protocol Summary: Five-year Follow-up] * [https://abcdstudy.org/wp-content/uploads/2019/12/flyer_protocol_MidYearSummary_eg.pdf ABCD Protocol Summary: Mid-year Follow-up] === ABCD Study Release Notes === The ABCD Study release notes were also used in the making of this page, and can be found below. Release notes were used as a second point of reference in determining when data were collected, data table names, and for citing measures. [https://wiki.abcdstudy.org/release-notes/start-page.html ABCD Study Release Notes] === ABCD Data Dictionary === The ABCD Data Dictionary was used to group measures, as well as to generate table, sub-scale, and variable names. The Data Dictionary was used as an additional point of reference in determining when data were collected. [https://data-dict.abcdstudy.org/? ABCD Data Dictionary] == Overview of Measures == {| class="wikitable sortable mw-collapsible" |+ABCD Core - Overview of Measures by Wave ''*Imaging data displayed separately below'' !Measure !Category !Subcategory !Source !Baseline !1-Year Follow-up !2-Year Follow-up !3-Year Follow-up !4-Year Follow-up !5-Year Follow-up !6-Year Follow-up !Mid-year Follow-up |- |Longitudinal Tracking |General Information |Administrative |Youth |Yes | | | | | | | |- | | | | | | | | | | | | |- |Latent Factors |General Information |Demographics |Youth | | | | | | | | |- | | | | | | | | | | | | |- |Occupation Survey |General Information |Demographics |Parent |No |No |Yes |Yes |Yes |Yes |No |No |- |PhenX Demographics Survey |General Information |Demographics |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Screener (Study Eligibility) |General Information |Administrative |Parent |Yes | | | | | | | |- | | | | | | | | | | | | |- |Blood - BASO, EOS, Hemoglobin, MCV, PLT/WBC/RBC counts, Immature Gran, Lymph, MCH, MCHC, MONO, MPV, NEUT, NRBC, RDW, Cholesterol, Burr Cells, Poikilocytosis, Hematocrit |Physical Health |Biospecimens |Youth |No |No |Yes |No |Yes |No |Yes |No |- |Blood (DNA) |Physical Health |Biospecimens |Youth |Yes* |No |Yes |No |Yes |No |Yes |No |- |Oral Fluids (pubertal hormones)<ref>Dolsen, E. A., Deardorff, J., & Harvey, A. G. (2019). Salivary Pubertal Hormones, Sleep Disturbance, and an Evening Circadian Preference in Adolescents: Risk Across Health Domains. ''The Journal of adolescent health : official publication of the Society for Adolescent Medicine'', ''64''(4), 523–529.</ref> |Physical Health |Biospecimens |Youth |Yes |Yes |Yes |Yes* |Yes |Yes* |Yes |No |- |Oral Fluids (DNA) |Physical Health |Biospecimens |Youth |Yes |No |No |No |No |No |No |No |- |[[COVID-19]] Annual Form |Physical Health |COVID |Youth |No |No |No |No |Yes |Yes |Yes |No |- |[[Blood pressure (OSCE)|Blood Pressure]] |Physical Health |Examination |Youth |No |No |Yes |No |Yes |No |Yes |No |- |PhenX Anthropometrics (height/weight/waist measurements)<ref>Centers for Disease Control (CDC; Division of Nutrition). (2016). Anthropometry Procedures Manual.</ref> |Physical Health |Examination |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Pain Questionnaire<ref>Luntamo, T., Sourander, A., Santalahti, P., Aromaa, M., & Helenius, H. (2012). Prevalence changes of pain, sleep problems and fatigue among 8-year-old children: years 1989, 1999, and 2005. ''Journal of pediatric psychology'', ''37''(3), 307–318.</ref> |Physical Health |Medical |Youth |No |No |Yes |Yes |Yes |No |Yes |No |- |Respiratory Functioning<ref>Gillman, M. W., & Blaisdell, C. J. (2018). Environmental influences on Child Health Outcomes, a Research Program of the National Institutes of Health. ''Current opinion in pediatrics'', ''30''(2), 260–262.</ref><ref>Asher, M. I., Keil, U., Anderson, H. R., Beasley, R., Crane, J., Martinez, F., Mitchell, E. A., Pearce, N., Sibbald, B., & Stewart, A. W. (1995). International Study of Asthma and Allergies in Childhood (ISAAC): rationale and methods. ''The European respiratory journal'', ''8''(3), 483–491.</ref> |Physical Health |Medical |Youth |No |No |No |No |Yes |No |Yes |No |- |Block Kids Food Screener - Youth<ref name=":1">Hunsberger, M., O’Malley, J., Block, T., & Norris, J. C. (2015). Relative validation of Block Kids Food Screener for dietary assessment in children and adolescents. ''Maternal & child nutrition'', ''11''(2), 260–270.</ref> |Physical Health |Nutrition |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Sports and Activities Involvement Questionnaire - Youth |Physical Health |Physical Activity |Youth |No |No |No |No |No |Yes |No |No |- |[[wikipedia:Youth_Risk_Behavior_Surveillance_System|Youth Risk Behavior Survey]] - Exercise |Physical Health |Physical Activity |Youth |Yes |No |Yes |Yes |Yes |No |Yes |No |- |Pubertal Development Scale and Menstrual Cycle Survey - Youth<ref name=":0">Petersen, A. C., Crockett, L., Richards, M., & Boxer, A. (1988). A self-report measure of pubertal status: Reliability, validity, and initial norms. ''Journal of youth and adolescence'', ''17''(2), 117–133.</ref> |Physical Health |Puberty |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[wikipedia:Munich_Chronotype_Questionnaire|Munich ChronoType Questionnaire]] (sleep)<ref>Zavada, A., Gordijn, M. C., Beersma, D. G., Daan, S., & Roenneberg, T. (2005). Comparison of the Munich Chronotype Questionnaire with the Horne-Ostberg’s Morningness-Eveningness Score. ''Chronobiology international'', ''22''(2), 267–278.</ref> |Physical Health |Sleep |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Medications |Physical Health | |Youth |No |No |No |No |No |No |Yes |No |- | | | | | | | | | | | | |- |Baby Teeth (substance and environmental toxin exposure)<ref>Cassidy-Bushrow, A. E., Wu, K. H., Sitarik, A. R., Park, S. K., Bielak, L. F., Austin, C., Gennings, C., Curtin, P., Johnson, C. C., & Arora, M. (2019). In utero metal exposures measured in deciduous teeth and birth outcomes in a racially-diverse urban cohort. ''Environmental research'', ''171'', 444–451.</ref> |Physical Health |Biospecimens |Parent |Yes |Yes |Yes |Yes* |Yes |Yes* |Yes |No |- |COVID-19 Annual Form |Physical Health |COVID |Parent |No |No |No |No |No |Yes |No |No |- |Breast Feeding Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref> |Physical Health |Development |Parent |No |No |No |Yes |No |No |No |No |- |Developmental History Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref><ref>Merikangas, K. R., Avenevoli, S., Costello, E. J., Koretz, D., & Kessler, R. C. (2009). National comorbidity survey replication adolescent supplement (NCS-A): I. Background and measures. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 367–379.</ref> |Physical Health |Development |Parent |Yes |No |No |No |Yes |No |No |No |- |Medical History Questionnaire<ref>Todd, R. D., Joyner, C. A., Heath, A. C., Neuman, R. J., & Reich, W. (2003). Reliability and stability of a semistructured DSM-IV interview designed for family studies. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''42''(12), 1460–1468.</ref> |Physical Health |Medical |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Ohio State [[wikipedia:Traumatic_brain_injury|TBI]] Screen - Short<ref>Bogner, J. A., Whiteneck, G. G., MacDonald, J., Juengst, S. B., Brown, A. W., Philippus, A. M., Marwitz, J. H., Lengenfelder, J., Mellick, D., Arenth, P., & Corrigan, J. D. (2017). Test-Retest Reliability of Traumatic Brain Injury Outcome Measures: A Traumatic Brain Injury Model Systems Study. ''The Journal of head trauma rehabilitation'', ''32''(5), E1–E16. </ref> |Physical Health |Medical |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |PhenX Medications Survey (Medications Inventory) |Physical Health |Medical |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Child Nutrition Assessment<ref>Morris, M. C., Tangney, C. C., Wang, Y., Sacks, F. M., Bennett, D. A., & Aggarwal, N. T. (2015). MIND diet associated with reduced incidence of Alzheimer’s disease. ''Alzheimer’s & dementia : the journal of the Alzheimer’s Association'', ''11''(9), 1007–1014.</ref> |Physical Health |Nutrition |Parent |No |Yes |No |No |No |No |No |No |- |Block Kids Food Screener - Parent<ref name=":1" /> |Physical Health |Nutrition |Parent |No |No |Yes |Yes |Yes |Yes |Yes |No |- |International Physical Activity Questionnaire<ref>Booth M. (2000). Assessment of physical activity: an international perspective. ''Research quarterly for exercise and sport'', ''71''(2 Suppl), S114–S120.</ref> |Physical Health |Physical Activity |Parent |No |No |No |Yes |No |No |No |No |- |Sports and Activities Involvement Questionnaire - Parent<ref>Huppertz, C., Bartels, M., de Zeeuw, E. L., van Beijsterveldt, C., Hudziak, J. J., Willemsen, G., Boomsma, D. I., & de Geus, E. (2016). Individual Differences in Exercise Behavior: Stability and Change in Genetic and Environmental Determinants From Age 7 to 18. ''Behavior genetics'', ''46''(5), 665–679.</ref> |Physical Health |Physical Activity |Parent |Yes |Yes |Yes |Yes |Yes |No |No |No |- |Pubertal Development Scale and Menstrual Cycle Survey - Parent<ref name=":0" /> |Physical Health |Puberty |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Sleep Disturbances Scale for Children<ref>Bruni, O., Ottaviano, S., Guidetti, V., Romoli, M., Innocenzi, M., Cortesi, F., & Giannotti, F. (1996). The Sleep Disturbance Scale for Children (SDSC). Construction and validation of an instrument to evaluate sleep disturbances in childhood and adolescence. ''Journal of sleep research'', ''5''(4), 251–261.</ref><ref>Ferreira, V. R., Carvalho, L. B., Ruotolo, F., de Morais, J. F., Prado, L. B., & Prado, G. F. (2009). Sleep disturbance scale for children: translation, cultural adaptation, and validation. ''Sleep medicine'', ''10''(4), 457–463.</ref> |Physical Health |Sleep |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- | | | | | | | | | | | | |- |Gender Identity (Youth)<ref name=":16">Potter, A., Dube, S., Allgaier, N., Loso, H., Ivanova, M., Barrios, L. C., Bookheimer, S., Chaarani, B., Dumas, J., Feldstein-Ewing, S., Freedman, E. G., Garavan, H., Hoffman, E., McGlade, E., Robin, L., & Johns, M. M. (2021). Early adolescent gender diversity and mental health in the Adolescent Brain Cognitive Development study. ''Journal of child psychology and psychiatry, and allied disciplines'', ''62''(2), 171–179.</ref><ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Reed, E., Salazar, M., Behar, A. I., Agah, N., Silverman, J. G., Minnis, A. M., Rusch, M., & Raj, A. (2019). Cyber Sexual Harassment: Prevalence and association with substance use, poor mental health, and STI history among sexually active adolescent girls. ''Journal of adolescence'', ''75'', 53–62.</ref> |Gender & Sexuality |Gender |Youth |No |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Sexual Behavior/Health<ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057. </ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref> |Gender & Sexuality |Sexuality |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- | | | | | | | | | | | | |- |Gender Identity (Parent)<ref name=":16" /><ref>Johnson, L. L., Bradley, S. J., Birkenfeld-Adams, A. S., Kuksis, M. A., Maing, D. M., Mitchell, J. N., & Zucker, K. J. (2004). A parent-report gender identity questionnaire for children. ''Archives of sexual behavior'', ''33''(2), 105–116.</ref><ref>Elizabeth, P. H., & Green, R. (1984). Childhood sex-role behaviors: similarities and differences in twins. ''Acta geneticae medicae et gemellologiae'', ''33''(2), 173–179.</ref> |Gender & Sexuality |Gender |Parent |No |Yes |Yes |Yes |No |No |No |No |- |Sexual Behavior/Health<ref name=":16" /><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref> |Gender & Sexuality |Sexuality |Parent |No |No |No |Yes* |Yes |Yes |Yes |No |- | | | | | | | | | | | | |- |Genetic Principal Components & Relatedness |Genetics |Genetics |Youth | | | | | | | | |- |Twin Zygosity Rating |Genetics |Genetics |Youth | | | | | | | | |- | | | | | | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Brief Problem Monitor Scale<ref>Achenbach, T. M. (2009). The Achenbach System of Empirically Based Assessment (ASEBA): Development, Findings, Theory, and Applications. Burlington, VT: University of Vermont Research Center for Children, Youth, & Families.</ref> |Mental Health |Broad Psychopathology |Youth |No |Yes |Yes |Yes |Yes |Yes |No |Yes |- |KSADS - Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Youth | | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Youth |No |No |Yes |No |Yes |No |Yes |No |- |Emotion Regulation Questionnaire<ref>Gross, J.J., & John, O.P. (2003). Individual differences in two emotion regulation processes: Implications for affect, relationships, and well-being. Journal of Personality and Social Psychology, 85, 348-362. </ref><ref>Garnefski, N., Rieffe, C., Jellesma, F., Terwogt, M. M., & Kraaij, V. (2007). Cognitive emotion regulation strategies and emotional problems in 9–11-year-old children: The development of an instrument. European Child & Adolescent Psychiatry, 16, 1–9.</ref><ref>Gullone, E., & Taffe, J. (2012). The Emotion Regulation Questionnaire for Children and Adolescents (ERQ-CA): a psychometric evaluation. Psychological assessment, ''24''(2), 409–417.</ref> |Mental Health |Emotion |Youth |No |No |No |No |No |Yes |No |No |- |NIH Toolbox Positive Affect Items<ref>Salsman, J. M., Butt, Z., Pilkonis, P. A., Cyranowski, J. M., Zill, N., Hendrie, H. C., Kupst, M. J., Kelly, M. A. R., Bode, R. K., Choi, S. W., Lai, J.-S. ., Griffith, J. W., Stoney, C. M., Brouwers, P., Knox, S. S., & Cella, D. (2013). Emotion assessment using the NIH Toolbox. ''Neurology'', ''80''(Issue 11, Supplement 3), S76–S86. <nowiki>https://doi.org/10.1212/wnl.0b013e3182872e11</nowiki> </ref><ref>{{Cite web|url=https://nihtoolbox.org/test/positive-affect/|title=Positive Affect|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref>Gershon, R.C., Wagster, M.V., et al., 2013. NIH toolbox for assessment of neurological and behavioral function. Neurology 80 (11 Suppl. 3), S2–6.</ref> |Mental Health |Emotion |Youth |No |Yes |No |Yes |No |Yes |No |Yes |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Youth |No |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[OToPS/Measures/7 Up 7 Down Inventory|7-Up Mania Items]]<ref name=":9">Youngstrom, E. A., Murray, G., Johnson, S. L., & Findling, R. L. (2013). The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory. ''Psychological Assessment'', ''25''(4), 1377–1383. <nowiki>https://doi.org/10.1037/a0033975</nowiki> </ref> |Mental Health |Mood |Youth |No |Yes |No |Yes |No |Yes |No |No |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |No |No |Yes |No |Yes |No |Yes |No |- |Peer Experiences Questionnaire<ref>De Los Reyes, A. & Prinstein, M. J. (2004). Applying depression-distortion hypotheses to the assessment of peer victimization in adolescents. Journal of Clinical Child and Adolescent Psychology, 33, 325-335.</ref><ref>Prinstein, M. J., Boergers, J., & Vernberg, E. M. (2001). Overt and relational aggression in adolescents: Social-psychological functioning of aggressors and victims. Journal of Clinical Child Psychology, 30, 477-489.</ref> |Mental Health |Peers |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Youth Resilience Scale<ref>Ungar, M., & Liebenberg, L. (2009). Cross-cultural consultation leading to the development of a valid measure of youth resilience: The International Resilience Project. ''Studia psychologica'', ''51''(2-3), 259-268.</ref> |Mental Health |Peers |Youth |Yes |No |Yes |Yes |Yes |Yes |Yes |No |- |Prodromal Psychosis Scale<ref>Karcher NR, Barch DM, Avenevoli S, Savill M, Huber RS, Simon TJ, Leckliter IN, Sher KJ, Loewy RL. Assessment of the Prodromal Questionnaire-Brief Child Version for Measurement of Self-reported Psychotic like Experiences in Childhood. JAMA Psychiatry. 2018 Aug 1;75(8):853-861.</ref><ref>Loewy, R.L., Bearden, C.E., et al., 2005. The prodromal questionnaire (PQ): preliminary validation of a self-report screening measure for prodromal and psychotic syndromes. Schizophr. Res. 79 (1), 117–125.</ref><ref>Ising, H.K., Veling, W., et al., 2012. The validity of the 16-item version of the Prodromal Questionnaire (PQ-16) to screen for ultra high risk of developing psychosis in the general help-seeking population. Schizophr. Bull. 38 (6), 1288–1296.</ref><ref>Therman, S., Lindgren, M., et al., 2014. Predicting psychosis and psychiatric hospital care among adolescent psychiatric patients with the Prodromal Questionnaire. Schizophr. Res. 158 (1–3), 7–10.</ref> |Mental Health |Psychosis |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[wikipedia:Kiddie_Schedule_for_Affective_Disorders_and_Schizophrenia|Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)]] Background Items Survey - Youth<ref name=":6">KAUFMAN, J., BIRMAHER, B., BRENT, D., RAO, U., FLYNN, C., MORECI, P., WILLIAMSON, D., & RYAN, N. (1997). Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data. ''Journal of the American Academy of Child & Adolescent Psychiatry'', ''36''(7), 980–988. <nowiki>https://doi.org/10.1097/00004583-199707000-00021</nowiki></ref><ref name=":7">Kaufman, J., Birmaher, B., Axelson, D., Perepletchikova, F., Brent, D., & Ryan, N. (2016). ''K-SADS-PL DSM-5''. <nowiki>https://pediatricbipolar.pitt.edu/sites/default/files/assets/Clinical%20tools/KSADS/KSADS_DSM_5_SCREEN_Final.pdf</nowiki> </ref><ref name=":2">Townsend, L, Kobak, K., Kearney, C., Milham, M., Andreotti, C., Escalera, J., Alexander, L., Gill, M.K., Birmaher, B., Sylvester, R., Rice, D., Deep, A., Kaufman, J. (2020).  Development of Three Web-Based Computerized Versions of the Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS-COMP) Child Psychiatric Diagnostic Interview: Preliminary Validity Data. Journal of the American Academy of Child and Adolescent Psychiatry, Feb;59(2):309-325. doi:10.1016/j.jaac. PMID: 31108163.</ref><ref name=":3">Kaufman, J., Kobak, K., Birmaher, B., & de Lacy, N. (2021). KSADS-COMP Perspectives on Child Psychiatric Diagnostic Assessment and Treatment Planning. Journal of the American Academy of Child and Adolescent Psychiatry, ''60''(5), 540–542.</ref> |Mental Health |Psychosocial |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems)<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Cyberbullying Questionnaire<ref>Stewart RW, Drescher CF, Maack DJ, Ebesutani C, Young J. The Development and Psychometric Investigation of the Cyberbullying Scale. J Interpers Violence. 2014 Aug;29(12):2218-2238. doi: 10.1177/0886260513517552. Epub 2014 Jan 14. PMID: 24424252.</ref> |Mental Health |Social |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Life Events Scale - Youth<ref name=":10">Tiet, Q.Q., Bird, H.R., et al., 2001. Relationship between specific adverse life events and psychiatric disorders. J. Abnorm. Child Psychol. 29 (2), 153–164.</ref><ref name=":15">Grant, K.E., Compas, B.E., et al., 2004. Stressors and child and adolescent psychopathology: measurement issues and prospective effects. J. Clin. Child Adolesc. Psychol. 33 (2), 412–425.</ref> |Mental Health |Stress |Youth |No |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[Behavioral Inhibition and Behavioral Activation System (BIS/BAS) Scales|PhenX Behavioral Inhibition/Behavioral Approach System (BIS/BAS) Scales]]<ref>Carver, C. & White, T. (1994). Behavioral Inhibition, Behavioral Activation, and affective response to impending reward and punishment: The BIS/BAS Scales. ''Journal of Personality and Social Psychology'', 67(2), 319-333.</ref><ref>Pagliaccio D, Luking KR, Anokhin AP, Gotlib IH, Hayden EP, Olino TM, Peng CZ, Hajcak G, Barch DM. Revising the BIS/BAS Scale to study development: Measurement invariance and normative effects of age and sex from childhood through adulthood. Psychol Assess. 2016 Apr;28(4):429-42. doi: 10.1037/pas0000186. Epub 2015 Aug 24. PMID: 26302106; PMCID: PMC4766059.</ref> |Mental Health |Temperament/Personality |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Urgency, Premeditation, Perseverance, Sensation Seeking, Positive Urgency, Impulsive Behavior (UPPS-P) for Children - Short Form (ABCD Version)<ref>Whiteside, S. P., & Lynam, D. R. (2001). The Five Factor Model and impulsivity: using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669-689. doi: 10.1016/S0191-8869(00)00064-7</ref><ref>Cyders, M. A., Smith, G. T., Spillane, N. S., Fischer, S., Annus, A. M., & Peterson, C. (2007). Integration of impulsivity and positive mood to predict risky behavior: Development and validation of a measure of positive urgency. ''Psychological Assessment, 19''(1), 107–118. <nowiki>https://doi.org/10.1037/1040-3590.19.1.107</nowiki></ref> |Mental Health |Temperament/Personality |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- | | | | | | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Agoraphobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Obsessive Compulsive Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Panic Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Separation Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |No |No |No |No |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Specific Phobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Autism Spectrum Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Autism Spectrum |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Behavior Checklist]]<ref name=":11">Achenbach, T. M., & Rescorla, L. A. (2003). Manual for the ASEBA adult forms & profiles. Research Center for Children, Youth, & Families, University of Vermont, Burlington, VT, USA.</ref> |Mental Health |Broad Psychopathology |Parent |No |No |Yes |No |Yes |No |No |No |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Self-Report]] (psychopathology)<ref name=":11" /> |Mental Health |Broad Psychopathology |Parent |Yes |No |Yes |No |Yes |No |No |No |- |[[wikipedia:Child_Behavior_Checklist|Child Behavior Checklist]]<ref>Achenbach TM, Rescorla LA. ''Manual for the ASEBA school-age forms & profiles: an integrated system of mult-informant assessment.'' Burlington: University of Vermont, Research Center for Children, Youth & Families; 2001.</ref> |Mental Health |Broad Psychopathology |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Family History Assessment Survey<ref>Brown SA, Brumback T, Tomlinson K, Cummins K, Thompson WK, Nagel BJ, De Bellis MD, Hooper SR, Clark DB, Chung T, Hasler BP, Colrain IM, Baker FC, Prouty D, Pfefferbaum A, Sullivan EV, Pohl KM, Rohlfing T, Nichols BN, Chu W, Tapert SF. The National Consortium on Alcohol and NeuroDevelopment in Adolescence (NCANDA): A Multisite Study of Adolescent Development and Substance Use. J Stud Alcohol Drugs. 2015 Nov;76(6):895-908. doi: 10.15288/jsad.2015.76.895. PMID: 26562597; PMCID: PMC4712659.</ref><ref>Rice JP, Reich T, Bucholz KK, Neuman RJ, Fishman R, Rochberg N, Hesselbrock VM, Nurnberger JI Jr, Schuckit MA, Begleiter H. Comparison of direct interview and family history diagnoses of alcohol dependence. Alcohol Clin Exp Res. 1995 Aug;19(4):1018-23. doi: 10.1111/j.1530-0277.1995.tb00983.x. PMID: 7485811.</ref> |Mental Health |Broad Psychopathology |Parent |Yes |No |No |No |No |No |Yes |No |- |KSADS Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Difficulty in Emotion Regulation Scale<ref>Bardeen, J. R., Fergus, T. A., Hannan, S. M., & Orcutt, H. K. (2016). Addressing psychometric limitations of the Difficulties in Emotion Regulation Scale through item modification. Journal of Personality Assessment.</ref><ref>Bunford, N., Dawson, A. E., Evans, S. W., Ray, A. R., Langberg, J. M., Owens, J. S., DuPaul, G. J., & Allan, D. M. (2020). The Difficulties in Emotion Regulation Scale-Parent Report: A Psychometric Investigation Examining Adolescents With and Without ADHD. Assessment, 27(5), 921–940.</ref> |Mental Health |Emotion |Parent |No |No |No |Yes |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (ADHD) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Oppositional Defiant Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |Yes |Yes |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Homicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |[[OToPS/Measures/7 Up 7 Down Inventory|General Behavior Inventory - Mania]]<ref name=":9" /> |Mental Health |Mood |Parent |Yes |Yes |Yes |No |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Parent <ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Psychotic Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosis |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |KSADS Background Items Survey - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosocial |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |Short Social Responsiveness Scale<ref>Aldridge, F. J., Gibbs, V. M., Schmidhofer, K., & Williams, M. (2012). Investigating the clinical usefulness of the Social Responsiveness Scale (SRS) in a tertiary level, autism spectrum disorder specific assessment clinic. ''Journal of autism and developmental disorders'', ''42''(2), 294–300.</ref><ref>Constantino J. N. (2011). The quantitative nature of autistic social impairment. ''Pediatric research'', ''69''(5 Pt 2), 55R–62R.</ref><ref>Constantino, J. N., Przybeck, T., Friesen, D., & Todd, R. D. (2000). Reciprocal social behavior in children with and without pervasive developmental disorders. ''Journal of developmental and behavioral pediatrics : JDBP'', ''21''(1), 2–11.</ref><ref>Constantino, J. N., & Todd, R. D. (2000). Genetic structure of reciprocal social behavior. ''The American journal of psychiatry'', ''157''(12), 2043–2045.</ref><ref>Constantino, J. N., & Todd, R. D. (2003). Autistic traits in the general population: a twin study. ''Archives of general psychiatry'', ''60''(5), 524–530.</ref><ref>Constantino, J. N., Gruber, C. P., Davis, S., Hayes, S., Passanante, N., & Przybeck, T. (2004). The factor structure of autistic traits. ''Journal of child psychology and psychiatry, and allied disciplines'', ''45''(4), 719–726.</ref><ref>Hus, V., Bishop, S., Gotham, K., Huerta, M., & Lord, C. (2013). Factors influencing scores on the social responsiveness scale. ''Journal of child psychology and psychiatry, and allied disciplines'', ''54''(2), 216–224.</ref><ref>Kaat, A. J., & Farmer, C. (2017). Commentary: Lingering questions about the Social Responsiveness Scale short form. A commentary on Sturm et al. (2017). ''Journal of child psychology and psychiatry, and allied disciplines'', ''58''(9), 1062–1064.</ref><ref>Norris, M., & Lecavalier, L. (2010). Screening accuracy of Level 2 autism spectrum disorder rating scales. A review of selected instruments. ''Autism : the international journal of research and practice'', ''14''(4), 263–284.</ref> |Mental Health |Social |Parent |No |Yes |No |No |No |Yes |No |No |- |KSADS Diagnostic Interview for DSM-5 (Post-Traumatic Stress Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Stress |Parent |Yes |No |Yes |No |Yes |No |Yes |No |- |Life Events Scale - Parent<ref name=":10" /><ref name=":15" /> |Mental Health |Stress |Parent |No |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[wikipedia:Perceived_Stress_Scale|Perceived Stress Scale]]<ref>Cohen, S., Kamarck, T., and Mermelstein, R. (1983). A global measure of perceived stress. Journal of Health and Social Behavior, 24, 386-396. </ref> |Mental Health |Stress |Parent |No |No |No |Yes |No |No |Yes |No |- |Early Adolescent Temperament Questionnaire<ref>Latham MD, Dudgeon P, Yap MBH, Simmons JG, Byrne ML, Schwartz OS, Ivie E, Whittle S, Allen NB. Factor Structure of the Early Adolescent Temperament Questionnaire-Revised. Assessment. 2020 Oct;27(7):1547-1561. doi: 10.1177/1073191119831789. Epub 2019 Feb 21. PMID: 30788984.</ref> |Mental Health |Temperament/Personality |Parent |No |No |Yes |No |No |No |No |No |- | | | | | | | | | | | | |- |Brief Problem Monitor |Mental Health |Broad Psychopathology |Teacher |Yes |Yes |Yes |Yes |No |Yes |No |No |- | | | | | | | | | | | | |- |[[wikipedia:Edinburgh_Handedness_Inventory|Edinburgh Handedness Inventory]] |Neurocognition |Administrative |Youth |Yes |No |No |No |No |No |No |No |- |Neurocognition Assessment Administration |Neurocognition |Administrative |Youth | | | | | | | | |- |[[wikipedia:Snellen_chart|Snellen Vision Screener]] |Neurocognition |Administrative |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Behavioral Indicator of Resiliency to Distress Task (BIRD)<ref>Lejuez, C. W., Kahler, C. W., & Brown, R. A. (2003). A modified computer version of the Paced Auditory Serial Addition Task (PASAT) as a laboratory-based stressor. ''The Behavior Therapist, 26''(4), 290–293.</ref><ref>Feldner, M. T., Leen-Feldner, E. W., Zvolensky, M. J., & Lejuez, C. W. (2006). Examining the association between rumination, negative affectivity, and negative affect induced by a paced auditory serial addition task. ''Journal of behavior therapy and experimental psychiatry'', ''37''(3), 171–187.</ref> |Neurocognition |Tasks |Youth |No |No |No |No |Yes |No |No |No |- |Cash Choice Task<ref>Wulfert, E., Block, J. A., Santa Ana, E., Rodriguez, M. L., & Colsman, M. (2002). Delay of gratification: impulsive choices and problem behaviors in early and late adolescence. ''Journal of personality'', ''70''(4), 533–552.</ref><ref>Anokhin, A. P., Golosheykin, S., Grant, J. D., & Heath, A. C. (2011). Heritability of delay discounting in adolescence: a longitudinal twin study. ''Behavior genetics'', ''41''(2), 175–183.</ref> |Neurocognition |Tasks |Youth |Yes |No |No |No |No |No |No |No |- |Delay Discounting Task<ref>Johnson, M. W., & Bickel, W. K. (2008). An algorithm for identifying nonsystematic delay-discounting data. ''Experimental and clinical psychopharmacology'', ''16''(3), 264–274.</ref> |Neurocognition |Tasks |Youth |No |Yes |No |Yes |No |Yes |No |No |- |Emotional Faces Stroop Task<ref>Başgöze, Z., Gönül, A. S., Baskak, B., & Gökçay, D. (2015). Valence-based Word-Face Stroop task reveals differential emotional interference in patients with major depression. ''Psychiatry research'', ''229''(3), 960–967.</ref><ref>Kane, M. J., & Engle, R. W. (2003). Working-memory capacity and the control of attention: the contributions of goal neglect, response competition, and task set to Stroop interference. ''Journal of experimental psychology. General'', ''132''(1), 47–70.</ref><ref>Stroop, J.R., 1935. Studies of interference in serial verbal reactions. J. Exp. Psychol. 18 (6), 643–662.</ref> |Neurocognition |Tasks |Youth |No |Yes |No |Yes |No |Yes |No |No |- |Game of Dice Task<ref>Brand, M., Fujiwara, E., Borsutzky, S., Kalbe, E., Kessler, J., & Markowitsch, H. J. (2005). Decision-making deficits of Korsakoff patients in a new gambling task with explicit rules: associations with executive functions. ''Neuropsychology'', ''19''(3), 267–277.</ref><ref>Drechsler, R., Rizzo, P., & Steinhausen, H. C. (2008). Decision-making on an explicit risk-taking task in preadolescents with attention-deficit/hyperactivity disorder. ''Journal of neural transmission (Vienna, Austria : 1996)'', ''115''(2), 201–209.</ref><ref>Duperrouzel, J. C., Hawes, S. W., Lopez-Quintero, C., Pacheco-Colón, I., Coxe, S., Hayes, T., & Gonzalez, R. (2019). Adolescent cannabis use and its associations with decision-making and episodic memory: Preliminary results from a longitudinal study. ''Neuropsychology'', ''33''(5), 701–710.</ref><ref>Ross, J. M., Graziano, P., Pacheco-Colón, I., Coxe, S., & Gonzalez, R. (2016). Decision-Making Does not Moderate the Association between Cannabis Use and Body Mass Index among Adolescent Cannabis Users. ''Journal of the International Neuropsychological Society : JINS'', ''22''(9), 944–949.</ref> |Neurocognition |Tasks |Youth |No |No |Yes |No |Yes |No |No |No |- |Little Man Task<ref>Acker, W. (1982). “A computerized approach to psychological screening—The Bexley-Maudsley Automated Psychological Screening and The Bexley-Maudsley Category Sorting Test.” ''International Journal of Man-Machine Studies'', ''17''(3): 361-369.</ref><ref>Nixon, S. J., Prather, R. A., & Lewis, B. (2014). Sex differences in alcohol-related neurobehavioral consequences. In Edith V. Sullivan and Adolf Pfefferbaum (Eds.), Alcohol and the nervous system (Handbook of clinical neurology, 3rd series (Vol. 125)). Oxford, United Kingdom, Elsevier, pp. 253-272.</ref> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - Dimensional Change Card Sort<ref>{{Cite web|url=https://nihtoolbox.org/test/dimensional-change-card-sort-test/|title=Dimensional Change Card Sort Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |No |No |No |No |Yes |No |- |NIH Toolbox Tasks - Flanker Inhibitory Control & Attention<ref>{{Cite web|url=https://nihtoolbox.org/test/flanker-inhibitory-control-and-attention-test-age-12/|title=Flanker Inhibitory Control and Attention Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - Oral Reading Recognition<ref>{{Cite web|url=https://nihtoolbox.org/test/a-dummy-iq-test/|title=Oral Reading Recognition Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - Pattern Comparison Processing Speed<ref>{{Cite web|url=https://nihtoolbox.org/test/pattern-comparison-processing-speed/|title=Pattern Comparison Processing Speed Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - Picture Sequence Memory<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-sequence-memory-test/|title=Picture Sequence Memory Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - Picture Vocabulary<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-vocabulary-test/|title=Picture Vocabulary Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12">McDonald, Skye (Ed.) (2014). Special series on the Cognition Battery of the NIH Toolbox. ''Journal of International Neuropsychological Society'', 20 (6), 487-651.</ref> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |NIH Toolbox Tasks - List Sorting Working Memory<ref>{{Cite web|url=https://nihtoolbox.org/test/test/|title=List Sorting Working Memory Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |Yes |No |No |No |Yes |No |Yes |No |- |RAVLT Delayed Recall<ref name=":13" /><ref name=":14" /> |Neurocognition | |Youth |Yes |No |Yes |No |No |No |No |No |- |Rey Auditory Verbal Leanring Task (RAVLT) <ref name=":13">Strauss, E., Sherman, E.M.S., & Spreen, O. (2006) A compendium of neuropsychological tests. Oxford University Press. New York, New York. Third Edition.</ref><ref name=":14">Lezak, M.D., Howieson, D.B., Bigler, E.D., & Tranel, D. (2012) Neuropsychological assessment. 5th Edition. Oxford University Press. New York, NY.</ref> |Neurocognition |Tasks |Youth |Yes |No |Yes |No |No |No |Yes |No |- |Social Influence Task<ref>Knoll, L. J., Leung, J. T., Foulkes, L., & Blakemore, S. J. (2017). Age-related differences in social influence on risk perception depend on the direction of influence. ''Journal of adolescence'', ''60'', 53–63.</ref> |Neurocognition |Tasks |Youth |No |No |Yes |No |Yes |No |No |No |- |Stanford Mental Arithmetic Response Time Evaluation (SMARTE)<ref>Starkey, G. S., & McCandliss, B. D. (2014). The emergence of “groupitizing” in children’s numerical cognition. ''Journal of experimental child psychology'', ''126'', 120–137.</ref> |Neurocognition |Tasks |Youth |No |No |No |Yes |No |Yes |No |No |- |Wechsler Intelligence Scale for Children - Matrix Reasoning Task<ref>Wechsler, D. (2014). Wechsler Intelligence Scale for Children - Fifth Edition Manual. San Antonio,TX, Pearson.</ref><ref>Daniel, M.H., Wahlstrom, D. & Zhang, O. (2014) Equivalence of Q-interactive® and Paper Administrations of Cognitive Tasks: WISC®–V: Q-Interactive Technical Report.</ref> |Neurocognition |Tasks |Youth |Yes |No |No |No |No |No |No |No |- | | | | | | | | | | | | |- |Barkley Deficits in Executive Functioning Scale<ref>Barkley RA (2010). Differential diagnosis of adults with ADHD: the role of executive function and self-regulation. ''J Clin Psychiatry'', 71(7), e17. doi: 10.4088/JCP.9066tx1c</ref><ref>Barkley RA (2011). ''Barkley deficits in executive functioning scale (BDEFS for adults)''. New York: Guilford Press.</ref><ref>Barkley RA (2012). ''Barkley Deficits in Executive Functioning Scale--Children and Adolescents (BDEFS-CA)'': Guilford Press.</ref> |Neurocognition |Questionnaire |Parent |No |No |No |No |No |No |Yes |No |- | | | | | | | | | | | | |- |[[wikipedia:Breathalyzer|Alcohol Toxicology]] |Substance Use |Drug Toxicology |Youth |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Hair Drug Toxicology<ref name=":41">Wade, N. E., Tapert, S. F., Lisdahl, K. M., Huestis, M. A., & Haist, F. (2022). Substance use onset in high-risk 9-13 year-olds in the ABCD study. Neurotoxicol Teratol, 91, 107090.</ref><ref name=":42">Wade, N. E., Sullivan, R. M., Tapert, S. F., Pelham, W. E., 3rd, Huestis, M. A., Lisdahl, K. M., & Haist, F. (2023). Concordance between substance use self-report and hair analysis in community-based adolescents. Am J Drug Alcohol Abuse, 49(1), 76-84.</ref> |Substance Use |Drug Toxicology |Youth |Yes |Yes |Yes |Yes* |Yes |Yes* |Yes |No |- |Nicotine Toxicology<ref name=":4">''NicAlert | NYMOX''. (2024). Nymox.com. <nowiki>https://nymox.com/products/nicalert</nowiki> </ref><ref name=":38">''Alere iScreen Dip Card''. (2024). Globalpointofcare.abbott. <nowiki>https://www.globalpointofcare.abbott/us/en/product-details/toxicology-iscreen.html</nowiki></ref><ref name=":39">''Reditest® Smoke Cassette | Test Devices | Redwood Toxicology Laboratory''. (2024). Redwoodtoxicology.com. <nowiki>https://www.redwoodtoxicology.com/devices/doa_redismoke</nowiki></ref><ref name=":40">''ReditestTM Smoke Cassette''. (2024). Toxicology.abbott. <nowiki>https://www.toxicology.abbott/us/en/products/reditest-smoke-cassette.html</nowiki></ref> |Substance Use |Drug Toxicology |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes |No |- |Saliva Drug Toxicology |Substance Use |Drug Toxicology |Youth |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |Yes |No |- |Urine Drug Toxicology<ref name=":38" /> |Substance Use |Drug Toxicology |Youth |No |No |No |No |Yes* |Yes* |Yes |No |- |Alcohol Expectancies Questionnaire (AEQ-AB)<ref>Brown, S. A., Christiansen, B. A., & Goldman, M. S. (1987). The Alcohol Expectancy Questionnaire: an instrument for the assessment of adolescent and adult alcohol expectancies. ''Journal of studies on alcohol'', ''48''(5), 483–491.</ref><ref>Greenbaum, P. E., Brown, E. C., & Friedman, R. M. (1995). Alcohol expectancies among adolescents with conduct disorder: prediction and mediation of drinking. ''Addictive behaviors'', ''20''(3), 321–333.</ref><ref>Stein, L. A., Katz, B., Colby, S. M., Barnett, N. P., Golembeske, C., Lebeau-Craven, R., & Monti, P. M. (2007). Validity and Reliability of the Alcohol Expectancy Questionnaire-Adolescent, Brief. ''Journal of child & adolescent substance abuse'', ''16''(2), 115–127.</ref> |Substance Use |SU Attitude |Youth |Yes |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |Alcohol Motives Questionnaire (PhenX)<ref>Cooper, M. L. (1994). Motivations for alcohol use among adolescents: Development and validation of a four-factor model. Psychological Assessment, 6, 117−128.</ref><ref>Grant, V. V., Stewart, S. H., O’Connor, R. M., Blackwell, E., & Conrod, P. J. (2007). Psychometric evaluation of the five-factor Modified Drinking Motives Questionnaire–Revised in undergraduates. ''Addictive behaviors'', ''32''(11), 2611–2632.</ref><ref>Kuntsche, E., & Kuntsche, S. (2009). Development and validation of the Drinking Motive Questionnaire Revised Short Form (DMQ-R SF). Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53, ''38''(6), 899–908.</ref> |Substance Use |SU Attitude |Youth |No |No |No |No |Yes* |No |Yes* |No |- |Cigarette Expectancies (ASCQ)<ref>Lewis-Esquerre, J. M., Rodrigue, J. R., & Kahler, C. W. (2005). Development and validation of an adolescent smoking Consequence questionnaire. ''Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco'', ''7''(1), 81–90.</ref> |Substance Use |SU Attitude |Youth |No |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |Electronic Nictotine Delivery Systems Expectancies Questionnaire<ref>Pokhrel, P., Lam, T.H., Pagano, I., Kawamoto, C.T., & Herzog, T.A. (2018). YPokhrel, P., Lam, T. H., Pagano, I., Kawamoto, C. T., & Herzog, T. A. (2018). Young adult e-cigarette use outcome expectancies: Validity of a revised scale and a short scale. ''Addictive behaviors'', ''78'', 193–199.</ref> |Substance Use |SU Attitude |Youth |No |No |No |Yes |No |Yes |No |No |- |Electronic Nicotine Delivery System Motives Inventory<ref>Centers for Disease Control (CDC; Division of Nutrition). (2016). Anthropometry Procedures Manual.</ref><ref name=":24">Piper, M. E., Piasecki, T. M., Federman, E. B., Bolt, D. M., Smith, S. S., Fiore, M. C., & Baker, T. B. (2004). A multiple motives approach to tobacco dependence: the Wisconsin Inventory of Smoking Dependence Motives (WISDM-68). Journal of consulting and clinical psychology, ''72''(2), 139–154.</ref> |Substance Use |SU Attitude |Youth |No |No |No |No |Yes* |No |Yes* |No |- |Marijuana Effect Expectancy Questionnaire (MEEQ-B)<ref name=":21">Torrealday, O., Stein, L. A., Barnett, N., Golembeske, C., Lebeau, R., Colby, S. M., & Monti, P. M. (2008). Validation of the Marijuana Effect Expectancy Questionnaire-Brief. ''Journal of child & adolescent substance abuse'', ''17''(4), 1–17.</ref> |Substance Use |SU Attitude |Youth |No |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |Marijuana Motives Questionnaire (PhenX)<ref>Lee, C. M., Neighbors, C., Hendershot, C. S., & Grossbard, J. R. (2009). Development and preliminary validation of a comprehensive marijuana motives questionnaire. Journal of studies on alcohol and drugs, ''70''(2), 279–287.</ref><ref>Simons, J., Correia, C. J., Carey, K. B., & Borsari, B. E. (1998). Validating a five-factor marijuana motives measure: Relations with use, problems, and alcohol motives. Journal of Counseling Psychology, ''45''(3), 265.</ref> |Substance Use |SU Attitude |Youth |No |No |No |No |Yes* |No |Yes* |No |- |PATH Intention to Use Alcohol, Nicotine, and Marijuana Survey <ref>Pierce, J. P., Choi, W. S., Gilpin, E. A., Farkas, A. J., & Merritt, R. K. (1996). Validation of susceptibility as a predictor of which adolescents take up smoking in the United States. Health psychology : official journal of the Division of Health Psychology, American Psychological Association, ''15''(5), 355–361.</ref><ref>Strong, D. R., Hartman, S. J., Nodora, J., Messer, K., James, L., White, M., Portnoy, D. B., Choiniere, C. J., Vullo, G. C., & Pierce, J. (2015). Predictive Validity of the Expanded Susceptibility to Smoke Index. Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, ''17''(7), 862–869.</ref> |Substance Use |SU Attitude |Youth |Yes* |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |PhenX Peer Group Deviance Survey<ref>Freedman, D., Thornton, A., Camburn, D., Alwin, D., & Young-demarco, L. (1988). The life history calendar: a technique for collecting retrospective data. Sociological methodology, ''18'', 37–68.</ref> |Substance Use |SU Attitude |Youth |Yes* |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |PhenX Peer Tolerance of Use<ref name=":19" /><ref name=":20" /> |Substance Use |SU Attitude |Youth |No |Yes* |Yes* |Yes |Yes |Yes |No |No |- |PhenX Perceived Harm of Substance Use<ref name=":19">Johnston, Lloyd D.; O’Malley, P. M.; Bachman, J. G.; Schulenberg, J. E.. (2009). Monitoring the Future. National Results on Adolescent Drug Use: Overview of Key Findings, 2009. NIH Publication Number 10-7583</ref><ref name=":20">Miech, R. A.; Johnston, L. D.; O’Malley, P. M.; Bachman, J. G.; Schulenberg, J. E.. (2015). Monitoring the Future National Survey Results on Drug Use, 1975-2014. Volume 1, Secondary School Students. Ann Arbor: Institute for Social Research: The University of Michigan.</ref> |Substance Use |SU Attitude |Youth |Yes |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |Reasons for Electronic Nicotine Delivery Systems Use<ref name=":24" /><ref>Wills, T. A., Sandy, J. M., & Yaeger, A. M. (2002). Moderators of the relation between substance use level and problems: test of a self-regulation model in middle adolescence. Journal of abnormal psychology, ''111''(1), 3–21.</ref> |Substance Use |SU Attitude |Youth |No |No |No |No |Yes* |No |Yes* |No |- |Tobacco Motives Inventory<ref>Smith, S. S., Piper, M. E., Bolt, D. M., Fiore, M. C., Wetter, D. W., Cinciripini, P. M., & Baker, T. B. (2010). Development of the Brief Wisconsin Inventory of Smoking Dependence Motives. ''Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco'', ''12''(5), 489–499.</ref> |Substance Use |SU Attitude |Youth |No |No |No |No |Yes* |No |Yes* |No |- |Vaping Expectancies<ref name=":21" /> |Substance Use |SU Attitude |Youth |No |No |No |Yes* |No |Yes |No |No |- |Vaping Motives<ref>Diez, S. L., Cristello, J. V., Dillon, F. R., De La Rosa, M., & Trucco, E. M. (2019). Validation of the electronic cigarette attitudes survey (ECAS) for youth. Addictive behaviors, ''91'', 216–221.</ref> |Substance Use |SU Attitude |Youth |No |No |Yes* |No |Yes* |No |Yes* |No |- |Alcohol Hangover Symptoms (HSS)<ref>Slutske, W. S., Piasecki, T. M., & Hunt-Carter, E. E. (2003). Development and initial validation of the Hangover Symptoms Scale: prevalence and correlates of Hangover Symptoms in college students. Alcoholism, clinical and experimental research, ''27''(9), 1442–1450.</ref> |Substance Use |SU Consequence |Youth |Yes |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Alcohol Subjective Effects (SRE; PhenX)<ref>Schuckit, M. A., Smith, T. L., & Tipp, J. E. (1997). The Self-Rating of the Effects of alcohol (SRE) form as a retrospective measure of the risk for alcoholism. Addiction (Abingdon, England), ''92''(8), 979–988.</ref> |Substance Use |SU Consequence |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Cannabis Withdrawal Scale (CWS)<ref>Allsop, D. J., Norberg, M. M., Copeland, J., Fu, S., & Budney, A. J. (2011). The Cannabis Withdrawal Scale development: patterns and predictors of cannabis withdrawal and distress. Drug and alcohol dependence, ''119''(1-2), 123–129.</ref> |Substance Use |SU Consequence |Youth |No |No |No |Yes* |Yes* |Yes* |Yes* |No |- |Drug Problem Index (DAPI)<ref name=":23">Johnson, V., & White, H. R. (1989). An investigation of factors related to intoxicated driving behaviors among youth. Journal of studies on alcohol, ''50''(4), 320–330.</ref><ref>Caldwell, P. E. (2002). Drinking levels, related problems and readiness to change in a college sample. Alcoholism Treatment Quarterly, ''20''(2), 1-15.</ref><ref>Kingston, J., Clarke, S., Ritchie, T., & Remington, B. (2011). Developing and validating the “composite measure of problem behaviors”. Journal of clinical psychology, ''67''(7), 736–751.</ref> |Substance Use |SU Consequence |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |KSADS - Alcohol/Drug Use Disorder - Youth<ref name=":3" /> |Substance Use |SU Consequence |Youth |No |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Marijuana Problem Index (MAPI)<ref name=":23" /><ref>Zvolensky, M. J., Vujanovic, A. A., Bernstein, A., Bonn-Miller, M. O., Marshall, E. C., & Leyro, T. M. (2007). Marijuana use motives: A confirmatory test and evaluation among young adult marijuana users. Addictive behaviors, ''32''(12), 3122–3130.</ref> |Substance Use |SU Consequence |Youth |Yes |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Marijuana Subjective Effects<ref>Agrawal, A., Madden, P. A., Bucholz, K. K., Heath, A. C., & Lynskey, M. T. (2014). Initial reactions to tobacco and cannabis smoking: a twin study. Addiction (Abingdon, England), ''109''(4), 663–671.</ref><ref>Agrawal, A., Madden, P. A., Martin, N. G., & Lynskey, M. T. (2013). Do early experiences with cannabis vary in cigarette smokers?. ''Drug and alcohol dependence'', ''128''(3), 255–259.</ref> |Substance Use |SU Consequence |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Nicotine Dependence (PATH)<ref name=":22">Pomerleau, O. F., Pomerleau, C. S., & Namenek, R. J. (1998). Early experiences with tobacco among women smokers, ex-smokers, and never-smokers. Addiction (Abingdon, England), ''93''(4), 595–599.</ref><ref>Prokhorov, A. V., Pallonen, U. E., Fava, J. L., Ding, L., & Niaura, R. (1996). Measuring nicotine dependence among high-risk adolescent smokers. Addictive behaviors, ''21''(1), 117–127.</ref> |Substance Use |SU Consequence |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Nicotine Subjective Response<ref name=":22" /><ref>Rodriguez, D., & Audrain-McGovern, J. (2004). Construct validity analysis of the early smoking experience questionnaire for adolescents. Addictive behaviors, ''29''(5), 1053–1057.</ref> |Substance Use |SU Consequence |Youth |No |Yes |Yes |Yes |Yes |Yes |Yes |No |- |[[Evidence-based assessment/Rx4DxTx of SubstanceUse|Rutgers Alcohol Problem Index (RAPI)]]<ref>White, H. R., & Labouvie, E. W. (1989). Towards the assessment of adolescent problem drinking. Journal of studies on alcohol, ''50''(1), 30–37.</ref> |Substance Use |SU Consequence |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |PhenX Community Risk and Protective Factors<ref name=":25">Arthur, M. W., Briney, J. S., Hawkins, J. D., Abbott, R. D., Brooke-Weiss, B. L., & Catalano, R. F. (2007). Measuring risk and protection in communities using the Communities That Care Youth Survey. Evaluation and program planning, ''30''(2), 197–211.</ref><ref name=":26">Trentacosta, C. J., Criss, M. M., Shaw, D. S., Lacourse, E., Hyde, L. W., & Dishion, T. J. (2011). Antecedents and outcomes of joint trajectories of mother-son conflict and warmth during middle childhood and adolescence. Child development, ''82''(5), 1676–1690.</ref> |Substance Use |SU Environment |Youth |No |No |Yes* |Yes |Yes |Yes |Yes |No |- |Sibling Use<ref>Samek, D. R., Goodman, R. J., Riley, L., McGue, M., & Iacono, W. G. (2018). The Developmental Unfolding of Sibling Influences on Alcohol Use over Time. Journal of youth and adolescence, ''47''(2), 349–368.</ref> |Substance Use |SU Environment |Youth |No |No |No |Yes |Yes |Yes |Yes |No |- |Caffeine Intake Survey - Substance Use Interview <ref name=":17" /><ref name=":18">Jackson, K. M., Barnett, N. P., Colby, S. M., & Rogers, M. L. (2015). The prospective association between sipping alcohol by the sixth grade and later substance use. Journal of studies on alcohol and drugs, ''76''(2), 212–221.</ref> |Substance Use |Substance Use |Youth |Yes* |Yes* |Yes* |Yes |Yes |Yes |Yes |No |- |ISay II Q2 Sipping Items (sip) - Substance Use Interview<ref name=":17" /><ref name=":18" /> |Substance Use |Substance Use |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Low Level Marijuana Use (puff/taste) - Substance Use Interview<ref name=":17" /><ref name=":18" /> |Substance Use |Substance Use |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Low Level Tobacco Use (puff) - Substance Use Interview<ref name=":17" /><ref name=":18" /> |Substance Use |Substance Use |Youth |No |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Participant Last Use Survey (PLUS) (Day 1/2/3/4) - Youth<ref name=":17">Lisdahl, K. M., Sher, K. J., Conway, K. P., Gonzalez, R., Feldstein Ewing, S. W., Nixon, S. J., Tapert, S., Bartsch, H., Goldstein, R. Z., & Heitzeg, M. (2018). Adolescent brain cognitive development (ABCD) study: Overview of substance use assessment methods. Developmental cognitive neuroscience, ''32'', 80–96.</ref> |Substance Use |Substance Use |Youth |Yes* |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Substance Use Phone Interview<ref name=":17" /> |Substance Use |Substance Use |Youth |No |No |No |No |No |No |No |Yes |- |Timeline Follow-Back Survey<ref name=":17" /><ref>Sobell, L. C., & Sobell, M. B. (1996). Time Line Follow Back. User s Guide, Toronto. ''Addiction Research Foundation''.</ref> |Substance Use |Substance Use |Youth |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |Yes* |No |- |Opportunity to Use Questionnaire |Substance Use | |Youth |No |No |No |No |No |Yes |Yes |No |- | | | | | | | | | | | | |- |KSADS - Alcohol/Drug Use Disorder - Parent<ref name=":3" /> |Substance Use |SU Consequence |Parent |Yes |No |Yes |Yes |Yes |Yes |Yes |No |- |Household Substance Use, Density, Storage & Second-Hand Exposure<ref>Bartels, K., Mayes, L. M., Dingmann, C., Bullard, K. J., Hopfer, C. J., & Binswanger, I. A. (2016). Opioid Use and Storage Patterns by Patients after Hospital Discharge following Surgery. ''PloS one'', ''11''(1), e0147972.</ref><ref>Friese, B., Grube, J. W., & Moore, R. S. (2012). How parents of adolescents store and monitor alcohol in the home. ''The journal of primary prevention'', ''33''(2-3), 79–83.</ref> |Substance Use |SU Environment |Parent |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Parent Rules Survey<ref>Dishion, T.J., Kavanagh, K., 2003. Intervening in Adolescent Problem Behavior: A Family-centered Approach. The Guilford Press, New York, NY.</ref><ref>Dishion, T. J., Nelson, S. E., & Kavanagh, K. (2003). The family check-up with high-risk young adolescents: Preventing early-onset substance use by parent monitoring. Behavior Therapy, ''34''(4), 553-571.</ref><ref name=":18" /><ref>Jackson, K. M., Roberts, M. E., Colby, S. M., Barnett, N. P., Abar, C. C., & Merrill, J. E. (2014). Willingness to drink as a function of peer offers and peer norms in early adolescence. ''Journal of studies on alcohol and drugs'', ''75''(3), 404–414.</ref> |Substance Use |SU Environment |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |PhenX Community Risk and Protective Factors<ref name=":25" /><ref name=":26" /> |Substance Use |SU Environment |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Participant Last Use Survey (PLUS) (Day 1/2/3/4) - Parent<ref name=":17" /> |Substance Use |Substance Use |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- | | | | | | | | | | | | |- |Wills Problem Solving Scale<ref>Wills, T. A., Ainette, M. G., Stoolmiller, M., Gibbons, F. X., & Shinar, O. (2008). Good self-control as a buffering agent for adolescent substance use: an investigation in early adolescence with time-varying covariates. ''Psychology of addictive behaviors : journal of the Society of Psychologists in Addictive Behaviors'', ''22''(4), 459–471.</ref> |Culture & Environment |Cognition |Youth |No |Yes |No |Yes |No |No |No |No |- |Perceived Discrimination Scale<ref>Garnett, B. R., Masyn, K. E., Austin, S. B., Miller, M., Williams, D. R., & Viswanath, K. (2014). The intersectionality of discrimination attributes and bullying among youth: an applied latent class analysis. Journal of youth and adolescence, 43(8), 1225–1239.</ref><ref>Phinney, J. S., Madden, T., & Santos, L. J. (1998). Psychological variables as predictors of perceived ethnic discrimination among minority and immigrant adolescents. ''Journal of Applied Social Psychology, 28''(11), 937–953</ref> |Culture & Environment |Community |Youth |No |Yes |Yes |No |Yes |No |Yes |No |- |PhenX Neighborhood Safety/Crime Survey - Youth<ref name=":33">Mujahid, M. S., Diez Roux, A. V., Morenoff, J. D., & Raghunathan, T. (2007). Assessing the measurement properties of neighborhood scales: from psychometrics to ecometrics. ''American journal of epidemiology'', ''165''(8), 858–867.</ref> |Culture & Environment |Community |Youth |Yes |Yes |Yes |No |Yes |No |No |No |- |Mexican American Cultural Values Scale - Youth<ref>Knight, G.P., Gonzales, N.A., Saenz, D.S., Bonds, D.D., German, M., Deardorff, J., Roosav, M.W., Updegraff, K.A., 2010. The Mexican American cultural values scale for adolescents and adults. J. Early Adolesc. 30 (3), 444–481.</ref> |Culture & Environment |Culture |Youth |No |No |Yes |Yes |Yes |No |Yes |No |- |Multi-Group Ethnic Identity - Revised - Youth<ref name=":36">Phinney, J. S., & Ong, A. D. (2007). Conceptualization and measurement of ethnic identity: Current status and future directions. Journal of Counseling Psychology, ''54''(3), 271-281.</ref> |Culture & Environment |Culture |Youth |No |No |No |Yes |No |Yes |Yes |No |- |Native American Acculturation Survey - Youth<ref name=":37">Garrett MT, Pichette EF. Red as an apple: Native American acculturation and counseling with or without reservation. Journal of Counseling and Development. 2000;78:3–13. </ref> |Culture & Environment |Culture |Youth |No |No |No |No |No |No |Yes* |No |- |PhenX Acculturation Survey - Youth<ref name=":28">Alegria, M., Takeuchi, D., Canino, G., Duan, N., Shrout, P., Meng, X.-L., Gong, F., et al. (2004). Considering context, place, and culture: the national Latino and Asian American study. Int. J. Methods Psychiatr. Res. 13 (4), 208–22.</ref><ref name=":29">Marin, G., F. Sabogal, B. V. Marin, R. Otero-Sabogal and E. J. Perez-Stable (1987). “Development of a Short Acculturation Scale for Hispanics.” Hispanic Journal of Behavioral Sciences 9(2): 183-205.</ref> |Culture & Environment |Culture |Youth |Yes |Yes |Yes |Yes |Yes |No |No |No |- |Vancouver Index of Acculturation - Youth<ref name=":35">Ryder, A.G., Alden, L.E., Paulhus, D.L., 2000. Is acculturation unidimensional or bidimensional? A head-to-head comparison in the prediction of personality, self-identity, and adjustment. J. Pers. Soc. Psychol. 79 (1), 49–65.</ref> |Culture & Environment |Culture |Youth |No |No |No |Yes |No |Yes |No |No |- |Pet Ownership<ref>Purweal, R., Christley, R., Kordas, K., Joinson, C., Meints, K., Gee, N., & Westgarth, C. (2017). Companion animals and child/adolescent development: A systematic review of the evidence. International Journal of Environmental Research and Public Health, 14(3), 234-259.</ref> |Culture & Environment |Family |Youth |No |No |No |Yes |No |No |No |No |- |PhenX [[wikipedia:Family_Environment_Scale|Family Environment Scale]] - Family Conflict - Youth<ref name=":32">Moos, R.H., Moos, B.S. (1994). Family Environment Scale Manual. Consulting Psychologists Press, Palo Alto, CA.</ref> |Culture & Environment |Family |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Acceptance Subscale from Children's Report of Parental Behavior Inventory (CRPBI) - Short<ref>Schaefer, E.S., 1965. A configurational analysis of children’s reports of parent behavior. J. Consult. Psychol. 29, 552–557.</ref><ref>Schludermann, E. H., & Schludermann, S. M. (1988). Children’s Report on Parent Behavior (CRPBI-108, CRPBI-30) for older children and adolescents. Winnipeg, MB, Canada: University of Manitoba.</ref><ref>Barber, B. K., Olsen, J. E., & Shagle, S. C. (1994). Associations between parental psychological and behavioral control and youth internalized and externalized behaviors. Child development, 65(4), 1120-1136.</ref><ref>Barber, B. K., & Olsen, J. A. (1997). Socialization in context: Connection, regulation, and autonomy in the family, school, and neighborhood, and with peers. Journal of Adolescent Research, 12(2), 287-315</ref> |Culture & Environment |Parenting |Youth |Yes |Yes |No |Yes |Yes |Yes |Yes |No |- |Multidimensional Neglectful Behavior Scale<ref>Dubowitz, H., Villodas, M. T., Litrownik, A. J., Pitts, S. C., Hussey, J. M., Thompson, R., … & Runyan, D. (2011). Psychometric properties of a youth self-report measure of neglectful behavior by parents. Child Abuse & Neglect, 35(6), 414-424.</ref> |Culture & Environment |Parenting |Youth |No |No |No |Yes |No |No |Yes |No |- |Parental Monitoring Survey<ref>Chilcoat, H. D., & Anthony, J. C. (1996). Impact of parent monitoring on initiation of drug use through late childhood. Journal of the American Academy of Child and Adolescent Psychiatry, ''35''(1), 91–100.</ref><ref name=":30">Karoly, H. C., Callahan, T., Schmiege, S. J., & Feldstein Ewing, S. W. (2015). Evaluating the Hispanic Paradox in the context of adolescent risky sexual behavior: the role of parent monitoring. Journal of pediatric psychology, 41(4), 429-440.</ref><ref name=":31">Stattin, H., & Kerr, M. (2000). Parental monitoring: a reinterpretation. Child development, ''71''(4), 1072–1085.</ref> |Culture & Environment |Parenting |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Peer Behavior Profile: Prosocial Peer Involvement & Delinquent Peer Involvement<ref>Bingham, C. R., Fitzgerald, H. E., & Zucker, R. A. (1995). Peer Behavior Profile/Peer Activities Questionnaire. Unpublished questionnaire. Department of Psychology, Michigan State University. East Lansing.</ref><ref>Hirschi, T. (1969). Causes of delinquency. Berkeley, CA: University of California Press.</ref><ref>Jessor, R., & Jessor, S.L. (1977). Problem behavior and psychosocial development: A longitudinal study of youth. New York, Academic Press.</ref> |Culture & Environment |Peers |Youth |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Peer Network Health: Protective Scale<ref>Mason, M., Light, J., Campbell, L., Keyser-Marcus, L., Crewe, S., Way, T., Saunders, H., King, L., Zaharakis, N.M., & McHenry, C. (2015). Peer network counseling with urban adolescents: A randomized controlled trial with moderate substance users. Journal of Substance Abuse Treatment, 58, 16-24.</ref> |Culture & Environment |Peers |Youth |No |No |Yes |Yes |Yes |No |Yes |No |- |Resistance to Peer Influence Scale/Questionnaire<ref>Steinberg, L., & Monahan, K. C. (2007). Age differences in resistance to peer influence. ''Developmental psychology'', ''43''(6), 1531–1543.</ref> |Culture & Environment |Peers |Youth |No |No |No |No |Yes |Yes |Yes |No |- |PhenX School Risk & Protective Factors Survey<ref>Arthur, M. W., Briney, J. S., Hawkins, J. D., Abbott, R. D., Brooke-Weiss, B. L., & Catalano, R. F. (2007). Measuring risk and protection in communities using the Communities That Care Youth Survey. ''Evaluation and program planning'', ''30''(2), 197–211.</ref><ref>Hamilton, C. M., Strader, L. C., Pratt, J. G., Maiese, D., Hendershot, T., Kwok, R. K., Hammond, J. A., Huggins, W., Jackman, D., Pan, H., Nettles, D. S., Beaty, T. H., Farrer, L. A., Kraft, P., Marazita, M. L., Ordovas, J. M., Pato, C. N., Spitz, M. R., Wagener, D., Williams, M., … Haines, J. (2011). The PhenX Toolkit: get the most from your measures. American journal of epidemiology, ''174''(3), 253–260.</ref> |Culture & Environment |School |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |School Attendance of Youth & Grades<ref name=":34">Zucker RA, Gonzalez R, Feldstein Ewing SW, Paulus MP, Arroyo J, Fuligni A, Morris AS, Sanchez M, Wills T. Assessment of culture and environment in the Adolescent Brain and Cognitive Development Study: Rationale, description of measures, and early data. Dev Cogn Neurosci. 2018 Aug;32:107-120</ref> |Culture & Environment |School |Youth |No |No |Yes |Yes |Yes |No |Yes |No |- |Prosocial Behavior Survey - Youth<ref name=":27">Goodman, R., Meltzer, H., Bailey, V., 1998. The strengths and difficulties questionnaire: a pilot study on the validity of the self-report version. Eur. Child Adolesc. Psychiatry 7(3), 125–130.</ref> |Culture & Environment |Temperament/ |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Activity Space |Culture & Environment | |Youth |No |No |No |No |No |Yes |Yes |No |- | | | | | | | | | | | | |- |Community Cohesion (PhenX)<ref>National Archive of Criminal Justice Data (NACJD), Project on Human Development in Chicago Neighborhoods (PHDCN). Community Survey 1994-1995.</ref><ref>PhenX Protocol - Neighborhood Collective Efficacy - Community Cohesion and Informal Social Control.</ref> |Culture & Environment |Community |Parent |No |No |Yes |No |Yes |No |Yes |No |- |PhenX Neighborhood Safety/Crime Survey - Parent<ref>Echeverria, S. E., Diez-Roux, A. V., et al. (2004) Reliability of self-reported neighborhood characteristics. J Urban Health 81(4): 682-701.</ref><ref name=":33" /> |Culture & Environment |Community |Parent |Yes |Yes |Yes |No |Yes |Yes |No |No |- |Mexican American Cultural Values Scale<ref name=":36" /> |Culture & Environment |Culture |Parent |Yes |Yes |Yes |Yes |Yes |No |Yes |No |- |Multi-Group Ethnic Identity Survey<ref name=":36" /> |Culture & Environment |Culture |Parent |Yes |No |Yes |Yes |Yes |Yes |Yes |No |- |Native American Acculturation Scale<ref name=":37" /> |Culture & Environment |Culture |Parent |Yes |No |No |No |No |No |No |No |- |PhenX Acculturation Survey - Parent<ref name=":28" /><ref name=":29" /> |Culture & Environment |Culture |Parent |Yes |Yes |Yes |Yes |Yes |No |No |No |- |Vancouver Index of Acculturation - Parent<ref name=":35" /> |Culture & Environment |Culture |Parent |Yes |No |Yes |Yes |Yes |Yes |Yes |No |- |PhenX Family Environment Scale - Family Conflict - Parent<ref name=":32" /><ref>Sanford, K., Bingham, C.R., & Zucker, R.A. (1999). Validity Issues with the Family Environment Scale: Psychometric Resolution and Research Application with Alcoholic Families. Psychological Assessment, 11(3),315‑325.</ref> |Culture & Environment |Family |Parent |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Parental Monitoring Survey<ref name=":30" /><ref name=":31" /> |Culture & Environment |Parenting |Parent |No |No |No |No |No |Yes |No |No |- |School Attendance of Youth & Grades<ref name=":34" /> |Culture & Environment |School |Parent |No |No |Yes |Yes |Yes |Yes |Yes |No |- |Prosocial Behavior Survey - Parent<ref name=":27" /> |Culture & Environment |Temperament/Personality |Parent |Yes |Yes |Yes |Yes |Yes |No |Yes |No |- |HOME Short Form Cognitive Stimulation<ref>Bailey, C.T. & Boykin, A.W. (2001). The role of task variability and home contextual factors in the academic performance and task motivation of African American elementary school children. ''The Journal of Negro Education, 70''(1/2), 84-95. <nowiki>http://www.jstor.org/stable/2696285</nowiki></ref><ref>Boykin, A.W. & Cunningham, R.T. The effects of movement expressiveness in story content and learning context on the analogical reasoning performance of African American Children. ''Negro Education, 70''(1/2), 72-83. <nowiki>http://www.jstor.org/stable/2696284</nowiki></ref><ref>Zaslow, M. J., Weinfield, N. S., Gallagher, M., Hair, E. C., Ogawa, J. R., Egeland, B., ... & De Temple, J. M. (2006). Longitudinal prediction of child outcomes from differing measures of parenting in a low-income sample. ''Developmental Psychology'', ''42''(1), 27-37. <nowiki>https://doi.org/10.1037/0012-1649.42.1.27</nowiki></ref> |Culture & Environment | |Parent |No |No |No |No |No |Yes |No |No |- |Driving |Culture & Environment | |Parent |No |No |No |No |No |No |Yes |No |- | | | | | | | | | | | | |- |Fitbit - Physical Activity (Daily) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Fitbit - Physical Activity (Weekly) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Fitbit - Sleep (Daily) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Fitbit - Sleep (Weekly) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |EARS - Device Usage Statistics (Pilot) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |EARS - Device Usage Statistics |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |EARS - Post-Assessment Survey |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |EARS/Vibrent - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- |Screen Time Questionnaire |Novel Technologies |Actigraphy |Youth |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |- |Vibrent - Device Usage Statistics (Pilot) |Novel Technologies |Actigraphy |Youth | | | | | | | | |- | | | | | | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Parent | | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Parent | | | | | | | | |- |EARS - Post-Assessment Survey |Novel Technologies |Screen Use |Parent | | | | | | | | |- |EARS - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Screen Use |Parent | | | | | | | | |- |Parent Screen Time Questionnaire |Novel Technologies |Screen Use |Parent |Yes |Yes |No |Yes |No |Yes |No |No |- | | | | | | | | | | | | |- |School Records |Other |School Records |Other |Yes |Yes |Yes |Yes |Yes |Yes |Yes |No |} {| class="wikitable sortable mw-collapsible" |+ABCD Core - Imaging Data !Measure !Category !Subcategory !Source !Baseline !1-Year Follow-up !2-Year Follow-up !3-Year Follow-up !4-Year Follow-up !5-Year Follow-up !6-Year Follow-up !Mid-Year Follow-up |- |MRI Info |Brain Imaging |Administrative |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Pre/Post-Scan Questionnaires |Brain Imaging |Administrative |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Scanning Checklist and Notes |Brain Imaging |Administrative |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Fractional Anisotropy - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity (Atlas Track) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Longitudinal Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Mean Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Transverse Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Free Normalized Isotropic - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Directional - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Hindered Normalized Total - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Directional - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Isotropic - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Restricted Normalized Total - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Correlations (Gordon Network to Subcortical) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Correlations (Gordon Network) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Temporal Variance (Desikan) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Temporal Variance (Destrieux) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Temporal Variance (Gordon) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Resting State fMRI - Temporal Variance (Subcortical) |Brain Imaging |Resting State fMRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Cortical Thickness (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Cortical Thickness (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Cortical Thickness (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Sulcal Depth (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Sulcal Depth (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Sulcal Depth (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Surface Area (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Surface Area (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Surface Area (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity (Subcortical) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray Matter (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray/White Contrast (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray/White Contrast (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - Gray/White Contrast (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - White Matter (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - White Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T1 Intensity - White Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity (Subcortical) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray Matter (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray/White Contrast (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray/White Contrast (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - Gray/White Contrast (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - White Matter (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - White Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |T2 Intensity - White Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (Desikan) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (Destrieux) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Volume (Subcortical) |Brain Imaging |Structural MRI |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref name=":43">Knutson B, Westdorp A, Kaiser E, Hommer D (2000) FMRI visualization of brain activity during a monetary incentive delay task. NeuroImage 12: 20–27.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small loss (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small loss (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small loss (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small reward (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small reward (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated large vs. small reward (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Anticipated small reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Behavioral Performance |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Loss positive vs. negative feedback (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Loss positive vs. negative feedback (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Loss positive vs. negative feedback (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Post-Scan Questionnaire |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Reward positive vs. negative feedback (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Reward positive vs. negative feedback (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Reward positive vs. negative feedback (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Recognition memory behavioral performance |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44">Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |0-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |0-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |0-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back vs. 0-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":52">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back vs. 0-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":53">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |2-back vs. 0-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":54">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Behavioral performance |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref name=":45">Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Emotion vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Face vs. place (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Face vs. place (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Face vs. place (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Negative face vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Negative face vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Negative face vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Place (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Place (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Place (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Positive face vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Positive face vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Positive face vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Behavioral performance |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46">Logan GD (1994) On the ability to inhibit thought and action: A users’ guide to the stop signal paradigm. In D. Dagenbach & T. H. Carr (Eds), Inhibitory processes in attention, memory, and language: 189-239. San Diego: Academic Press</ref> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Any stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Any stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Any stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct go vs. fixation (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct go vs. fixation (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct go vs. fixation (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. incorrect stop (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. incorrect stop (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Correct stop vs. incorrect stop (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. incorrect stop (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. incorrect stop (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect go vs. incorrect stop (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Incorrect stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Automatic - Post-processing |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |MRI Clinical Report/FIndings |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Manual - Freesurfer |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Manual - Post-processing - Diffusion MRI |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Manual - Post-processing - Functional MRI |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Manual - Post-processing - Structural MRI - T2w |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Motion |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Diffusion MRI |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Event |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Resting State fMRI |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Structural MRI - T1 |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Structural MRI - T2 |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Task fMRI - All |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Task fMRI - MID |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Task fMRI - N-Back |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Raw - Task fMRI - SST |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |- |Recommended Image Inclusion |Brain Imaging |Quality Control |Youth |Yes |No |Yes |No |Yes |No |Yes |No |} {| class="wikitable sortable mw-collapsible" |+ABCD Core - Linked Data !Measure !Category !Subcategory !Source !Data Collection Information !Associated Date |- |Geocoding from Residential History |Linked Data |Administrative |Other |Collected at yearly follow-ups from baseline. |Yearly follow-ups. |- |Satellite-based NO2 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |Residential history derived avg/max of NO2 in 2016 at residential address at 1x1km2 |2016 |- |Satellite-based O3 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |Residential history derived avg/max of O3 in 2016 at residential address at 1x1km2 |2016 |- |Satellite-based PM2.5 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |Residential history derived avg/max of PM2.5 in 2016 at residential address at 1x1km2 |2016 |- |Satellite-based Particulate Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |Residential history-derived annual average of Br, Ca, Cu, C, Fe, K, NH4+, Ni, NO3-, Pb, Si, SO2-4, V, Zn in ng/m^3 at 50m. | |- |Satellite-based Pollution Measures for Prenatal Addresses |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |9 month average of PM2.5, NO2, O3 exposure at current address #1 of birth year. | |- |Neighborhood SES and Demographics |Linked Data |Amenities & Services |NaDA |Residential history derived demographic information, ACS 2013-2017, census tract; NaNDA. | |- |Parks |Linked Data |Amenities & Services |NaDA |Residential history derived information on park type and number by county and census tract; NaNDA. | |- |Performing Arts and Sports Recreation Orgs |Linked Data |Amenities & Services |NaDA |Residential history derived information on performing arts and sports recreation orgs by county and census tract; NaNDA. | |- |Religious/Civic Organizations |Linked Data |Amenities & Services |NaDA |Residential history derived information on number of religious, civic, and social informations by county and census tract; NaNDA. | |- |Social Service |Linked Data |Amenities & Services |NaDA |Residential history derived information on the number of social services by county and census tract; NaNDA. | |- |Building Density |Linked Data |Built Environment |EPA |Residential history derived gross residential density. | |- |Crime |Linked Data |Built Environment |ICPSR |Residential history derived uniform crime reports by type of crime. | |- |Lead Risk |Linked Data |Built Environment |Vox |Percentage of individuals below -125 percent of poverty level, estimated percentage of homes at risk for lead-based paint exposure, and estimated lead risk in census tract. | |- |Population Density |Linked Data |Built Environment |EPA |Residential history derived UN adjusted population density. | |- |Road Proximity |Linked Data |Built Environment |Kalibrate |Residential history derived proximity to major roads. | |- |Traffic Density |Linked Data |Built Environment |Kalibrate |Average annual daily traffic counts at current address. | |- |Urban/Rural Area |Linked Data |Built Environment |Census |Census Tract Urban Classification at current address. | |- |Vehicle Density |Linked Data |Built Environment |ACS |Residential history derived aggregate number of vehicles per individual and sq. mile of land area. | |- |Walkability |Linked Data |Built Environment |EPA |Residential history derived national walkability index. | |- |Behavioral Health Measures |Linked Data |Community Health Burden |PLACES |Residential history derived health markers (measured via prevalence) among adults. | |- |Elevation of Address |Linked Data |Meteorology & Exposures |Google API |Residential history derived elevation. | |- |Estimates of Environmental Noise |Linked Data |Meteorology & Exposures |Harvard |Residential history derived anthropogenic and total sound level. | |- |Selected EJScreen Measures |Linked Data |Meteorology & Exposures |EJScreen |Residential history derived NATA air toxics cancer risk, respiratory hazard index, Diesel PM in micrograms of PM2.5 per cubic meter in air. | |- |Temperature Estimates |Linked Data |Meteorology & Exposures |PRISM |Residential history derived maximum temperature in degrees C at participant's residence at 5, 4, 3, 2, 1, 0 days prior to date of study visit (t-5 to t-0). | |- |VPD Estimates |Linked Data |Meteorology & Exposures |PRISM |Residential history derived maximum vapor pressure deficit in hPa at participant's address at 6, 5, 4, 3, 2, 1, 0 days prior to date of study visit (t-6 to t-0). | |- |Land-use Measures |Linked Data |Natural Space & Satellite |NLT |Built-up/cropland/grass land use, normalized difference built-up/vegetation/water index, night light radiance, population density, water area. |2017 |- |Measure of Land Cover and Tree Canopy |Linked Data |Natural Space & Satellite |NLCD |Residential history derived data on what percentage of the census tract is classified as developed (open and high/med/low intensity), barren land, forest (type specified), shrub/scrub, herbaceous, pasture, cultivated crop, wetland, tree canopy. |2016 |- |Alcohol Outlet Density |Linked Data |Neighborhood Social Factors |Census 2016 |Estimate of alcohol outlet density from ZIP Code Business Patterns |2016 |- |Anomie/Disenfranchisement/Social Capital |Linked Data |Neighborhood Social Factors |Census Return |Residential history derived mail return rate for the 2010 census and self-response rate for the 2014-2018 ACD per census tract. |2010, 2016-2018 |- |Number of Jobs and Job Density |Linked Data |Neighborhood Social Factors |LODES |Residential history derived total number of jobs, number of jobs by race and ethnicity, job density per sq. mile of land area by race and ethnicity | |- |Opportunity Zones and Investment Scores (OZ) |Linked Data |Neighborhood Social Factors |Other |Residential history derived opportunity zone designation and investment score. | |- |Rent and Mortgage Statistics |Linked Data |Neighborhood Social Factors |ACS |Residential history derived percent homeownership, median house value, percent rent burden, median rent per month. |2014-2018 |- |Social Mobility |Linked Data |Neighborhood Social Factors |Opportunity Atlas |Residential history derived opportunity atlas mean outcome and household income rank for children based on parent income (percentile of the national income distribution). |2014-2015 |- |Area Deprivation Index (ADI) |Linked Data |Neighborhood Composite Measures |Other |Residential history derived area deprivation index measured by income, education, occupation, rent/mortgage, crowding, amenity access, and car/phone ownership. | |- |Child Opportunity Index 2.0 (COI) |Linked Data |Neighborhood Composite Measures |Other |Child Opportunity Levels across education, health/environment, and social/economic domains (normed based on national, state, and metro area data). ''Education domain'': AP course enrollment, adult educational attainment, college enrollment, early childhood education centers/enrollment, high school graduation rate, third grade math/reading proficiency, school poverty, teacher experience. ''Health and Environment domain'': access to healthy food, green space, extreme heat exposure, health insurance coverage, ozone concentration, airborne microparticles, housing vacancy rate, walkability, hazardous waste dump sites, industrial pollutant exposure. ''Social and Economic domain'': poverty rate, public assistance rate, homeownership rate, high-skill employment, median household income, employment rate, commute, single-headed households. | |- |Minority Health Social Vulnerability Index (MHSVI) |Linked Data |Neighborhood Composite Measures |Other |Residential history derived demographic makeup of population (race/ethnicity/language), access to healthcare services, health insurance, computer/internet access. | |- |Social Vulnerability Index (SVI) |Linked Data |Neighborhood Composite Measures |CDC |Residential history derived census tract CDC SVI (total and percentile data based on demographic factors). |2014-2018 (5 year average). |- |Affordable Care Act Medicaid Expansion Data |Linked Data |Policy Vars |KFF |Was ACA expansion effective at date of baseline visit? |Baseline visit |- |CDC Opioid Prescription Dispensing Data per 100k Residents |Linked Data |Policy Vars |CDC |Residential history derived opioid prescription per 100K by state at baseline visit year, and 1-5 years before baseline visit year. |0-5 years before baseline visit year. |- |Cannabis Legalizations Categories by State |Linked Data |Policy Vars |NCSL and MPP |Marijuana state law during the same year as the assessment | |- |Gender Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |State-level indicators of sexism from survey and implicit bias measures, and of sexual orientation from structural variables. | |- |Immigration Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |State-level indictors of immigrant bias from survey and implicit bias measures and state-level structural variables. | |- |OPTIC-Vetted Co-prescribing Naloxone Policy Data |Linked Data |Policy Vars |OPTIC |Was any state co-prescribing nalaxone policy effective for all patients at the date of the baseline visit? |Baseline visit |- |OPTIC-Vetted Good Samaritan Policy Data |Linked Data |Policy Vars |OPTIC |Was any type of Good Samaritan law, specifically one providing protection from arrest or controlled substance possession, effective at the date of the baseline visit? |Baseline visit |- |OPTIC-Vetted Medical Marijuana Policy Data |Linked Data |Policy Vars |OPTIC |Residential history derived information on medical marijuana laws, recreational marijuana laws, dispensary legality, rec stores, and high CBD/low THC laws. |Baseline visit |- |OPTIC-Vetted Naloxone Policy Data |Linked Data |Policy Vars |OPTIC |Residential history derived data on Naloxone laws, including the presence of any Naloxone laws, legality of distribution through a standing or protocol order, and the legality of pharmacists prescribing Naloxone. |Baseline visit |- |OPTIC-Vetted Prescription Drug Monitoring Program Policy Data (PDMP) |Linked Data |Policy Vars |OPTIC |Residential history derived information on the presence of legislation requiring prescribers to access PDMP before prescribing, PDMP enabling legislation, and accessible "modern PDMP systems" at the date of baseline visit. |Baseline visit |- |Race Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |State level indicators of racism from survey and implicit bias measures and state-level structural variables. | |- |Dissimilarity Index |Linked Data |Residential Segregation |ACS |Residential history derived Black-White, Asian-White, Hispanic-White, and NonWhite-White metro-level dissimilarity indexes. |2014-2018 |- |Exposure/Interaction Index |Linked Data |Residential Segregation |ACS |Residential history derived Black-White, Asian-White, Hispanic-White, and NonWhite-White exposure/interaction indexes. |2014-2018 |- |Getis-Ord GI* Statistics |Linked Data |Residential Segregation |ICPSR |Residential history derived local GI* statistics (White, Black, Asian, and Hispanic rook and queen neighborhood tracts). | |- |Index of Concentration at the Extremes |Linked Data |Residential Segregation |ACS |Residential history derived index of concentration at the extremes. ''Income:'' (Households with an income of ≥ $100k - Households with an income of ≤ $25k)/total population. ''Income + Race:'' (Non-Hispanic white households with an income of ≥ $100k - Non-Hispanic Black households with an income of ≤ $25k)/total population. |2014-2018 |- |Multi-Group Entropy Index |Linked Data |Residential Segregation |ACS |Residential history derived multigroup entropy score (census tract/metro level) and index. |2014-2018 |- |County |Linked Data |School (Demographics) |SEDA |County-level statistics for school segregation and relative diversity, median income, BAPLUS rate, racial and ethnic make up, % of ELL students, % students receiving free and reduced lunch, poverty rate, % of rural/suburban/town/urban schools, single mother HH rate, SNAP reciept rate, total enrollment (grades 3-8), and unemployment rate. |2005-2009 and 2014-2018 averages |- |District |Linked Data |School (Demographics) |SEDA |Geo District-level statistics for BAAND rate, highest/lowest grade offered in district, school segregation and relative diversity, racial and ethnic make up, % ELL students, % students receiving free and reduced lunch, poverty rate, % of Special Ed students, SES composite, single mother HH rate, SNAP receipt rate, % of rural/suburban/town/urban schools, district enrollment (grades 3-8), and unemployment rate. |2005-2009 and 2014-2018 averages |- |Metro Area |Linked Data |School (Demographics) |SEDA |Metro area-level statistics for BAAND rate, school segregation and relative diversity, median income, county change between 2009 and 2013, racial and ethnic make up, % student receiving free and reduced lunch, % ECD in the metro, poverty rate, % of rural/suburban/town/urban schools, SES composite, single mother HH rate, SNAP receipt rate, total enrollment (grades 3-8), and unemployment rate. Information on if the metro area has data in the ACS and CCD. Metropolitan and micropolitan area definitions. |2005-2009 and 2014-2018 averages |- |School |Linked Data |School (Demographics) |SEDA | | |- |Commuting Zone |Linked Data |School (Math & Reading) |SEDA | | |- |County |Linked Data |School (Math & Reading) |SEDA | | |- |District |Linked Data |School (Math & Reading) |SEDA | | |- |Metro Area |Linked Data |School (Math & Reading) |SEDA | | |- |School |Linked Data |School (Math & Reading) |SEDA | | |- |Commuting Zone |Linked Data |School (Math Poolsub) |SEDA | | |- |County |Linked Data |School (Math Poolsub) |SEDA | | |- |District |Linked Data |School (Math Poolsub) |SEDA | | |- |Metro Area |Linked Data |School (Math Poolsub) |SEDA | | |- |Commuting Zone |Linked Data |School (Reading Poolsub) |SEDA | | |- |County |Linked Data |School (Reading Poolsub) |SEDA | | |- |District |Linked Data |School (Reading Poolsub) |SEDA | | |- |Metro Area |Linked Data |School (Reading Poolsub) |SEDA | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Substudy - Measures by Wave !Measure !Category !Subcategory !Source !Baseline !1-Year Follow-up !2-Year Follow-up !3-Year Follow-up !4-Year Follow-up !5-Year Follow-up !6-Year Follow-up !Mid-Year Follow-up |- |COVID-19 Fitbit Physical Activity (Daily) |COVID-19 |Actigraphy |Youth | | | | | | | | |- |COVID-19 Fitbit Physical Activity (Weekly) |COVID-19 |Actigraphy |Youth | | | | | | | | |- |COVID-19 Fitbit Post-Assessment Survey |COVID-19 |Actigraphy |Youth | | | | | | | | |- |COVID-19 Fitbit Sleep (Daily) |COVID-19 |Actigraphy |Youth | | | | | | | | |- |COVID-19 Fitbit Sleep (Weekly) |COVID-19 |Actigraphy |Youth | | | | | | | | |- |COVID-19 Questionnaire |COVID-19 |COVID |Youth | | | | | | | | |- |Endocannabinoid Substudy |Endocannabinoid |SU Consequence |Youth | | | | | | | | |- |Hurricane Irma Experiences |Hurricane Irma |Questionnaire |Youth | | | | | | | | |- |Reported Delinquency<ref name=":8">Elliott DS, Ageton SS, Huizinga D, Knowles BA, Canter RJ. ''The prevalence and incidence of delinquent behavior: 1976–1980 (National Youth Survey Report No. 26)'' Behavioral Research Institute; Boulder, CO: 1983.</ref> |Social Development |Delinquency |Youth |No |Yes |Yes |Yes |No |No |No |No |- |Difficulties in Emotion Regulation |Social Development |Emotion |Youth | | | | | | | | |- |Firearms (YRBSS) |Social Development |Firearm Storage |Youth | | | | | | | | |- |Alabama Parenting Questionnaire |Social Development |Parenting |Youth | | | | | | | | |- |Peer Behavior |Social Development |Peers |Youth | | | | | | | | |- |Personality Disposition |Social Development |Temperament/Personality |Youth | | | | | | | | |- |Victimization |Social Development |Victimization |Youth | | | | | | | | |- | | | | | | | | | | | | |- |COVID-19 Questionnaire |COVID-19 |COVID |Parent | | | | | | | | |- |Hurricane Irma Experiences |Hurricane Irma |Questionnaire |Parent | | | | | | | | |- |Visit Type |Social Development |Administrative |Parent | | | | | | | | |- |Perception of Neighborhood Scale |Social Development |Community |Parent | | | | | | | | |- |Reported Delinquency |Social Development |Delinquency |Parent | | | | | | | | |- |Difficulties in Emotion Regulation |Social Development |Emotion |Parent | | | | | | | | |- |Firearms (BRFSS) |Social Development |Firearm Storage |Parent | | | | | | | | |- |Alabama Parenting Questionnaire |Social Development |Parenting |Parent | | | | | | | | |- |Personality Disposition |Social Development |Temperament/Personality |Parent | | | | | | | | |- |Victimization |Social Development |Victimization |Parent | | | | | | | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Substudy - Linked Data !Measure !Category !Subcategory !Source !Data Collection Information !Associated Date |- |Administrative Information |COVID-19 |Geocoded Data |Youth |Percentage of time spent as address 1-5. | |- |CDC Policy Surveilance |COVID-19 |Geocoded Data |CDC |Policies and dates of policy changes for general gathering bans, stay at home orders, public masking orders, bar operation, and restaurant operation due to COVID. | |- |Johns Hopkins University COVID-19 Prevalence |COVID-19 |Geocoded Data |JHU |County-level data collected 0-6 days before questionnaire dissemination date: * Cumulative case count * Population-normed cumulative case count (per 100k) * New case count * Population-normed new case count (per 100k) * Population-normed new case count (per 100k; 7-day rolling mean) * Cumulative death count * Population-normed cumulative death count (per 100k) * New death count * Population-normed new death count (per 100k) * Population-normed new death count (per 100k; 7-day rolling mean) |COVID-19 questionnaire dissemination date |- |SafeGraph Social Distancing Metrics |COVID-19 |Geocoded Data |SafeGraph |SafeGraph Device data collected by census block 0-6 days before questionnaire dissemination date: * Median distance traveled from home (meters) * Median distance traveled from home 1 year prior (meters) * Dwell time (minutes) * Dwell time 1 year prior (minutes) * Completely at home ratio * Completely at home ratio 1 year prior * Exhibiting full-time work behavior * Exhibiting full-time work behavior 1 year prior * Exhibiting part-time work behavior * Exhibiting part-time work behavior 1 year prior |COVID-19 questionnaire dissemination data |- |US Bureau of Labor Statistics (BLS) and Census Data |COVID-19 |Geocoded Data |BLS/Census |Monthly unemployment rate (not seasonally adjusted), monthly unemployment rate 1 year prior (not seasonally adjusted) | |} == Notes - Measures by Wave == ====== Physical Health ====== * All measures in the ABCD protocol were revised to eliminate the use of binary gender classifications in the fall of 2020 * It needs to be clarified whether the COVID-19 Annual Form (Physical Health) and the COVID-19 Questionnaire (COVID-19 Substudy) are the same. ====== Brain Imaging ====== * "Yes/No" for imaging timepoints should be double-checked as PDFs of measures by wave broadly indicated that all brain imaging tests were done at every other yearly follow-up, but the specific tests that populate the rows of the table were taken from the online data dictionary where no information about the frequency of data collection was given. ====== Biospecimens ====== * In response to the COVID-19 pandemic, the ABCD study pivoted to remote testing in March of 2020. This affected the 2-, 3-, and 4-year follow-up assessments. The switch to remote testing did not allow for the collection of biospecimens. * From NIMH Data Archive ** Genomic data (short name: genomics_sample03) was collected from the following sources and was reported in ABCD Releases 2.0, 3.0, and 4.0. This data needs to be reconciled with that reported in the protocol by wave, which reported collection of DNA via oral fluids at baseline and DNA via blood every other year starting at baseline. *** Whole blood; saliva, brain; urine; serum; plasma; CSF; IPS; Fibroblast; Neuronal Progenitor; skin biopsy; temporal cortex; lymphoblastoid cell line; semen; frontal cortex; parietal cortex; basal ganglia; placenta; hair; occipital visual cortex; cerebellum; spleen; stool; muscle; kidney; liver; heart; cord blood; nasal swab; DNA; RNA; breastmilk; buccal swab; oral cells; rectal swab; cervicovaginal swab ====== Mental Health ====== * Variables for the following diagnoses, obtained using the KSADS, are present in the data: Adjustment Disorder, Agoraphobia, Alcohol Use Disorder, Attention Deficit Hyperactivity Disorder, Autism Spectrum Disorders, Bipolar Disorders, Conduct Disorder, Disruptive Mood Dysregulation Disorder, Depressive Disorders, Drug Use Disorders, Eating Disorders, Generalized Anxiety Disorder, Obsessive Compulsive Disorder, Panic Disorder, Post-Traumatic Stress Disorder, Psychosis, Selective Mutism, Separation Anxiety, Social Anxiety, and Suicidality * A switch was made to the KSADS 2.0 at the 3-year follow-up. This version introduced several changes ** New diagnostic categories: schizophrenia, schizoaffective disorder, schizophreniform disorder, and autism spectrum disorder. ** Branching algorithm to streamline supplement items. ** Additional questions on the end of supplements that determine episode classifications. ** Medication information, corresponding with diagnosis. ** A more rigorous probe for hallucinations that requires hallucinatory-like experiences to occur during daily activity. ** Moved paranoid ideation questions from the psychotic disorders screen to the supplement to minimize false positives. ** Moved item about hypersexuality in the Bipolar Disorder screen to the supplement to minimize false positives. ** A screen item pertaining to the duration of OCD-like symptoms was added, the obsessive thoughts about sex item was removed from the screen, and the phrasing of the initial OCD probe item was changed to minimize false positives. ** Back button added. *Need clarification about whether there is any overlap between the following: Adult Self-Report Survey, ASEBA Adult Self-Report (psychopathology), ASEBA Adult Behavior Checklist *It needs to be clarified whether the "Difficulty in Emotion Regulation Scale" from the parent-report mental health section of the core study is the same as the "Difficulties in Emotion Regulation" from the emotion subsection of the social development substudy. ====== Neurocognition ====== * The Flanker Task was included under the NIH Toolbox on the PDFs of Measures by Wave but was listed as a separate cognitive task in the data dictionary. * The RAVLT Delayed Recall task was listed as distinct in the PDFs of Measures by Wave, but not in the online data dictionary. ====== Substance Use ====== * Between March of 2020 and December of 2021 remote and hybrid testing in response to the COVID-19 pandemic required participants to complete some measures on their own devices. Remote performance was monitored via Zoom by research associates when feasible, and youth were asked to find private places to complete substance use measures, but this was not always possible. This reduction in privacy may affect responses. * Discrepancies in when substance use measures were given to youth exist between the ABCD Study release notes and the ABCD Protocol by Wave fact sheets ** Alcohol Expectancies Questionnaire (AEQ-AB) *** Release notes: Annually from baseline to 3-year follow-up. Expected future assessment at 5-year and 7-year follow ups. *** Protocol by wave: Annually from 1-year follow-up. (protocol by wave is correct) ** Marijuana Expectancies (MEEQ-B) *** Release notes: Annually from baseline to 3-year follow-up. Expected future assessments in 5-year and 7-year follow-ups. *** Protocol by wave: Annually from 1-year follow-up. 1-3 year follow up, something weird with 4-year follow-up ** Vaping Motives *** Release notes: 2-year and 4-year follow-ups. Expected future assessment at 6-year follow-up. *** Protocol by wave: 4- and 6-year follow-ups. (protocol by wave is correct just 3 and 4 year follow ups) ** PhenX Peer Tolerance of Substance Use *** Release notes: Administered annually baseline through 4-year follow-up. *** Protocol by wave: Administered annually 1-year follow-up through 5-year follow-up. (1/2/3/4 follow up) ** Cigarette Expectancies (ASCQ) (1/2/3/4) *** Release notes: Baseline through 3-year follow-up, expected future assessments at 5-year and 7-year follow-up. *** Protocol by wave: Annually since 1-year follow-up **Community Risk and Protective Factors (Parent) (baseline/1/2/3/4) ***Release notes: Annually since 2-year follow-up ***Protocol by wave: Annually since baseline. **Substance Use Density, Storage, and Exposure (Parent) (2/3/4) ***Release notes: Annually since baseline ***Protocol by wave: Annually since 2-year follow-up. ====== Culture & Environment ====== * Discrepancies in culture & environment measures between the ABCD study release notes and the ABCD protocol by wave fact sheets: ** Pet Ownership (3/4) *** Release notes: 3-year follow-up and 4-year follow-up. *** Protocol by wave: 3-year follow-up only. ** School Attendance of Youth & Grades (2/3) *** Release notes: 2- and 3-year follow-up. *** Protocol by wave: 2-, 3-, 4-, and 6-year follow up. ** Parental Monitoring Survey (parent) (4) *** Release notes: 4-year follow-up *** Protocol by wave: 5-year follow-up ====== Core - Linked Data ====== * All information about linked data was taken from the [https://data-dict.abcdstudy.org/? ABCD online Data Dictionary] ** Source and measure abbreviation were noted the same way, making it hard to differentiate between the two as no definitions were given. All sources should be checked. *In "Lead Risk" the online data dictionary noted that information on the "Percentage of individuals below -125 percent of poverty level" was taken. This number should be double checked for accuracy. *All data collected through derived residential history contains variables for primary, secondary, and tertiary residential addresses. ====== Substudy - Linked Data ====== * All information about linked data was taken from the [https://data-dict.abcdstudy.org/? ABCD online Data Dictionary] * All data contains variables for addresses 1-5. * For the Johns Hopkins University COVID-19 Prevalence linked dataset and the SafeGraph Social Distancing Metrics dataset, it was not clear which questionnaire the variable labels were referring to when stating that geocoded data was collected "0-6 days before from questionnaire dissemination date." It may be assumed that this is in reference to the COVID-19 questionnaire, which was administered at unspecified follow-ups through the COVID-19 substudy. == Coding Information == {| class="wikitable sortable mw-collapsible" |+ABCD Core - Coding Information ''*Imaging data displayed separately below'' !Measure !Category !Subcategory !Source !Table Name (Current) !Table Name (NDA 4.0) !Subscale Information !SAS Code !SPSS Code !R Code |- |Longitudinal Tracking |General Information |Administrative |Youth |abct_y_lt |general; acspsw03; abcd_lt01 | | | | |- | | | | | | | | | | |- |Latent Factors |General Information |Demographics |Youth |abcd_y_lf |abcd_sss01 | | | | |- | | | | | | | | | | |- |Occupation Survey |General Information |Demographics |Parent |abcd_p_ocp |abcd_occsp01 | | | | |- |PhenX Demographics Survey |General Information |Demographics |Parent |abcd_p_demo |abcd_lpds01; acspsw03; pdem02 | | | | |- |Screener (Study Eligibility) |General Information |Administrative |Parent |abcd_p_screen |abcd_screen01 | | | | |- | | | | | | | | | | |- |Blood - BASO, EOS, Hemoglobin, MCV, PLT/WBC/RBC counts, Immature Gran, Lymph, MCH, MCHC, MONO, MPV, NEUT, NRBC, RDW, Cholesterol, Burr Cells, Poikilocytosis, Hematocrit |Physical Health |Biospecimens |Youth |ph_y_bld |abcd_ybd01 | | | | |- |Blood (DNA) |Physical Health |Biospecimens |Youth | | | | | | |- |Oral Fluids (pubertal hormones)<ref>Dolsen, E. A., Deardorff, J., & Harvey, A. G. (2019). Salivary Pubertal Hormones, Sleep Disturbance, and an Evening Circadian Preference in Adolescents: Risk Across Health Domains. ''The Journal of adolescent health : official publication of the Society for Adolescent Medicine'', ''64''(4), 523–529.</ref> |Physical Health |Biospecimens |Youth |ph_y_sal_horm |abcd_hsss01; sph01 | | | | |- |Oral Fluids (DNA) |Physical Health |Biospecimens |Youth | | | | | | |- |[[COVID-19]] Annual Form |Physical Health |COVID |Youth |ph_y_covid |N/A | | | | |- |[[Blood pressure (OSCE)|Blood Pressure]] |Physical Health |Examination |Youth |ph_y_bp |abcd_bp01 | | | | |- |PhenX Anthropometrics (height/weight/waist measurements)<ref>Centers for Disease Control (CDC; Division of Nutrition). (2016). Anthropometry Procedures Manual.</ref> |Physical Health |Examination |Youth |ph_y_anthro |abcd_ant01 | | | | |- |Pain Questionnaire<ref>Luntamo, T., Sourander, A., Santalahti, P., Aromaa, M., & Helenius, H. (2012). Prevalence changes of pain, sleep problems and fatigue among 8-year-old children: years 1989, 1999, and 2005. ''Journal of pediatric psychology'', ''37''(3), 307–318.</ref> |Physical Health |Medical |Youth |ph_y_pq |abcd_pq01 | | | | |- |Respiratory Functioning<ref>Gillman, M. W., & Blaisdell, C. J. (2018). Environmental influences on Child Health Outcomes, a Research Program of the National Institutes of Health. ''Current opinion in pediatrics'', ''30''(2), 260–262.</ref><ref>Asher, M. I., Keil, U., Anderson, H. R., Beasley, R., Crane, J., Martinez, F., Mitchell, E. A., Pearce, N., Sibbald, B., & Stewart, A. W. (1995). International Study of Asthma and Allergies in Childhood (ISAAC): rationale and methods. ''The European respiratory journal'', ''8''(3), 483–491.</ref> |Physical Health |Medical |Youth |ph_y_resp |N/A | | | | |- |Block Kids Food Screener - Youth<ref name=":1">Hunsberger, M., O’Malley, J., Block, T., & Norris, J. C. (2015). Relative validation of Block Kids Food Screener for dietary assessment in children and adolescents. ''Maternal & child nutrition'', ''11''(2), 260–270.</ref> |Physical Health |Nutrition |Youth |ph_y_bkfs |N/A | | | | |- |Sports and Activities Involvement Questionnaire - Youth |Physical Health |Physical Activity |Youth |ph_y_saiq |sports_activ_read_music01 | | | | |- |[[wikipedia:Youth_Risk_Behavior_Surveillance_System|Youth Risk Behavior Survey]] - Exercise |Physical Health |Physical Activity |Youth |ph_y_yrb |abcd_yrb01 | | | | |- |Pubertal Development Scale and Menstrual Cycle Survey - Youth<ref name=":0">Petersen, A. C., Crockett, L., Richards, M., & Boxer, A. (1988). A self-report measure of pubertal status: Reliability, validity, and initial norms. ''Journal of youth and adolescence'', ''17''(2), 117–133.</ref> |Physical Health |Puberty |Youth |ph_y_pds |abcd_ssphy01; abcd_ypdms01 | | | | |- |[[wikipedia:Munich_Chronotype_Questionnaire|Munich ChronoType Questionnaire]] (sleep)<ref>Zavada, A., Gordijn, M. C., Beersma, D. G., Daan, S., & Roenneberg, T. (2005). Comparison of the Munich Chronotype Questionnaire with the Horne-Ostberg’s Morningness-Eveningness Score. ''Chronobiology international'', ''22''(2), 267–278.</ref> |Physical Health |Sleep |Youth |ph_y_mctq |abcd_mcqc01 | | | | |- |Medications |Physical Health |Medications |Youth | | | | | | |- | | | | | | | | | | |- |Baby Teeth (substance and environmental toxin exposure)<ref>Cassidy-Bushrow, A. E., Wu, K. H., Sitarik, A. R., Park, S. K., Bielak, L. F., Austin, C., Gennings, C., Curtin, P., Johnson, C. C., & Arora, M. (2019). In utero metal exposures measured in deciduous teeth and birth outcomes in a racially-diverse urban cohort. ''Environmental research'', ''171'', 444–451.</ref> |Physical Health |Biospecimens |Parent |ph_p_teeth |bteeth01 | | | | |- |COVID-19 Annual Form |Physical Health |COVID |Parent |ph_p_covid |N/A | | | | |- |Breast Feeding Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref> |Physical Health |Development |Parent |ph_p_bfq |breast_feeding01 | | | | |- |Developmental History Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref><ref>Merikangas, K. R., Avenevoli, S., Costello, E. J., Koretz, D., & Kessler, R. C. (2009). National comorbidity survey replication adolescent supplement (NCS-A): I. Background and measures. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 367–379.</ref> |Physical Health |Development |Parent |ph_p_dhx |abcd_devhxss01; dhx01 | | | | |- |Medical History Questionnaire<ref>Todd, R. D., Joyner, C. A., Heath, A. C., Neuman, R. J., & Reich, W. (2003). Reliability and stability of a semistructured DSM-IV interview designed for family studies. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''42''(12), 1460–1468.</ref> |Physical Health |Medical |Parent |ph_p_mhx |abcd_lpmh01; abcd_lssmh01; abcd_medhxss01; abcd_mx01 | | | | |- |Ohio State [[wikipedia:Traumatic_brain_injury|TBI]] Screen - Short<ref>Bogner, J. A., Whiteneck, G. G., MacDonald, J., Juengst, S. B., Brown, A. W., Philippus, A. M., Marwitz, J. H., Lengenfelder, J., Mellick, D., Arenth, P., & Corrigan, J. D. (2017). Test-Retest Reliability of Traumatic Brain Injury Outcome Measures: A Traumatic Brain Injury Model Systems Study. ''The Journal of head trauma rehabilitation'', ''32''(5), E1–E16. </ref> |Physical Health |Medical |Parent |ph_p_otbi |abcd_lpohstbi01; abcd_lsstbi01; abcd_otbi01; abcd_tbi01 | | | | |- |PhenX Medications Survey (Medications Inventory) |Physical Health |Medical |Parent |ph_p_meds |medsy01 | | | | |- |Child Nutrition Assessment<ref>Morris, M. C., Tangney, C. C., Wang, Y., Sacks, F. M., Bennett, D. A., & Aggarwal, N. T. (2015). MIND diet associated with reduced incidence of Alzheimer’s disease. ''Alzheimer’s & dementia : the journal of the Alzheimer’s Association'', ''11''(9), 1007–1014.</ref> |Physical Health |Nutrition |Parent |ph_p_cna |abcd_cna01; abcd_ssphp01 | | | | |- |Block Kids Food Screener - Parent<ref name=":1" /> |Physical Health |Nutrition |Parent |ph_p_bkfs |abcd_bkfs01 | | | | |- |International Physical Activity Questionnaire<ref>Booth M. (2000). Assessment of physical activity: an international perspective. ''Research quarterly for exercise and sport'', ''71''(2 Suppl), S114–S120.</ref> |Physical Health |Physical Activity |Parent |ph_p_ipaq |internat_physical_activ01; | | | | |- |Sports and Activities Involvement Questionnaire - Parent<ref>Huppertz, C., Bartels, M., de Zeeuw, E. L., van Beijsterveldt, C., Hudziak, J. J., Willemsen, G., Boomsma, D. I., & de Geus, E. (2016). Individual Differences in Exercise Behavior: Stability and Change in Genetic and Environmental Determinants From Age 7 to 18. ''Behavior genetics'', ''46''(5), 665–679.</ref> |Physical Health |Physical Activity |Parent |ph_p_saiq |abcd_lpsaiq01; abcd_lsssa01; abcd_saiq02; abcd_spacss01; sports_activ_read_music_p01 | | | | |- |Pubertal Development Scale and Menstrual Cycle Survey - Parent<ref name=":0" /> |Physical Health |Puberty |Parent |ph_p_pds |abcd_ppdms01; abcd_ssphp01 | | | | |- |Sleep Disturbances Scale for Children<ref>Bruni, O., Ottaviano, S., Guidetti, V., Romoli, M., Innocenzi, M., Cortesi, F., & Giannotti, F. (1996). The Sleep Disturbance Scale for Children (SDSC). Construction and validation of an instrument to evaluate sleep disturbances in childhood and adolescence. ''Journal of sleep research'', ''5''(4), 251–261.</ref><ref>Ferreira, V. R., Carvalho, L. B., Ruotolo, F., de Morais, J. F., Prado, L. B., & Prado, G. F. (2009). Sleep disturbance scale for children: translation, cultural adaptation, and validation. ''Sleep medicine'', ''10''(4), 457–463.</ref> |Physical Health |Sleep |Parent |ph_p_sds |abcd_sds01; abcd_ssphp01 | | | | |- | | | | | | | | | | |- |Gender Identity (Youth)<ref name=":16">Potter, A., Dube, S., Allgaier, N., Loso, H., Ivanova, M., Barrios, L. C., Bookheimer, S., Chaarani, B., Dumas, J., Feldstein-Ewing, S., Freedman, E. G., Garavan, H., Hoffman, E., McGlade, E., Robin, L., & Johns, M. M. (2021). Early adolescent gender diversity and mental health in the Adolescent Brain Cognitive Development study. ''Journal of child psychology and psychiatry, and allied disciplines'', ''62''(2), 171–179.</ref><ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Reed, E., Salazar, M., Behar, A. I., Agah, N., Silverman, J. G., Minnis, A. M., Rusch, M., & Raj, A. (2019). Cyber Sexual Harassment: Prevalence and association with substance use, poor mental health, and STI history among sexually active adolescent girls. ''Journal of adolescence'', ''75'', 53–62.</ref> |Gender & Sexuality |Gender |Youth |gish_y_gi |abcd_ygi01; abcd_yksad01; | | | | |- |Sexual Behavior/Health<ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057. </ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref> |Gender & Sexuality |Sexuality |Youth |gish_y_sex |abcd_gish2y01; abcd_yksad01 | | | | |- | | | | | | | | | | |- |Gender Identity (Parent)<ref name=":16" /><ref>Johnson, L. L., Bradley, S. J., Birkenfeld-Adams, A. S., Kuksis, M. A., Maing, D. M., Mitchell, J. N., & Zucker, K. J. (2004). A parent-report gender identity questionnaire for children. ''Archives of sexual behavior'', ''33''(2), 105–116.</ref><ref>Elizabeth, P. H., & Green, R. (1984). Childhood sex-role behaviors: similarities and differences in twins. ''Acta geneticae medicae et gemellologiae'', ''33''(2), 173–179.</ref> |Gender & Sexuality |Gender |Parent |gish_p_gi |abcd_lpds01; abcd_lpksad01; abcd_pgi01; dibf01; pdem02 | | | | |- |Sexual Behavior/Health<ref name=":16" /><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref> |Gender & Sexuality |Sexuality |Parent |gish_p_sex |abcd_lpksad01; dibf01 | | | | |- | | | | | | | | | | |- |Genetic Principal Components & Relatedness |Genetics |Genetics |Youth |gen_y_pihat |acspsw03 | | | | |- |Twin Zygosity Rating |Genetics |Genetics |Youth |gen_y_zygrat |abcd_tztab01 | | | | |- | | | | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |my_y_ksads_gad |generaled_anx_disorder01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |my_y_ksads_sad |social_anxiety_disorder01 | | | | |- |Brief Problem Monitor Scale<ref>Achenbach, T. M. (2009). The Achenbach System of Empirically Based Assessment (ASEBA): Development, Findings, Theory, and Applications. Burlington, VT: University of Vermont Research Center for Children, Youth, & Families.</ref> |Mental Health |Broad Psychopathology |Youth |mh_y_bpm |abcd_bpm01; abcd_yssbpm01 | | | | |- |KSADS - Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Youth |mh_y_ksads_ss |abcd_ksad501; ksads2daic_use_only01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Youth |mh_y_ksads_ed |eating_disorders01 | | | | |- |Emotion Regulation Questionnaire<ref>Gross, J.J., & John, O.P. (2003). Individual differences in two emotion regulation processes: Implications for affect, relationships, and well-being. Journal of Personality and Social Psychology, 85, 348-362. </ref><ref>Garnefski, N., Rieffe, C., Jellesma, F., Terwogt, M. M., & Kraaij, V. (2007). Cognitive emotion regulation strategies and emotional problems in 9–11-year-old children: The development of an instrument. European Child & Adolescent Psychiatry, 16, 1–9.</ref><ref>Gullone, E., & Taffe, J. (2012). The Emotion Regulation Questionnaire for Children and Adolescents (ERQ-CA): a psychometric evaluation. Psychological assessment, ''24''(2), 409–417.</ref> |Mental Health |Emotion |Youth |mh_y_erq |abcd_mhy02; emotion_reg_erq_feelings01 | | | | |- |NIH Toolbox Positive Affect Items<ref>Salsman, J. M., Butt, Z., Pilkonis, P. A., Cyranowski, J. M., Zill, N., Hendrie, H. C., Kupst, M. J., Kelly, M. A. R., Bode, R. K., Choi, S. W., Lai, J.-S. ., Griffith, J. W., Stoney, C. M., Brouwers, P., Knox, S. S., & Cella, D. (2013). Emotion assessment using the NIH Toolbox. ''Neurology'', ''80''(Issue 11, Supplement 3), S76–S86. <nowiki>https://doi.org/10.1212/wnl.0b013e3182872e11</nowiki> </ref><ref>{{Cite web|url=https://nihtoolbox.org/test/positive-affect/|title=Positive Affect|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref>Gershon, R.C., Wagster, M.V., et al., 2013. NIH toolbox for assessment of neurological and behavioral function. Neurology 80 (11 Suppl. 3), S2–6.</ref> |Mental Health |Emotion |Youth |mh_y_poa |abcd_yssbpm01; abcd_ytbpai01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Youth |mh_y_ksads_cd |conduct_disorder01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Youth |mh_y_ksads_si |suicidality01 | | | | |- |[[OToPS/Measures/7 Up 7 Down Inventory|7-Up Mania Items]]<ref name=":9">Youngstrom, E. A., Murray, G., Johnson, S. L., & Findling, R. L. (2013). The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory. ''Psychological Assessment'', ''25''(4), 1377–1383. <nowiki>https://doi.org/10.1037/a0033975</nowiki> </ref> |Mental Health |Mood |Youth |mh_y_7up |abcd_mhy02; abcd_y7mi01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |mh_y_ksads_bip |bipolar_disorders01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |mh_y_ksads_dep |depressive_disorders01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |mh_y_ksads_dmdd |disruptive_mood_dysreg01 | | | | |- |Peer Experiences Questionnaire<ref>De Los Reyes, A. & Prinstein, M. J. (2004). Applying depression-distortion hypotheses to the assessment of peer victimization in adolescents. Journal of Clinical Child and Adolescent Psychology, 33, 325-335.</ref><ref>Prinstein, M. J., Boergers, J., & Vernberg, E. M. (2001). Overt and relational aggression in adolescents: Social-psychological functioning of aggressors and victims. Journal of Clinical Child Psychology, 30, 477-489.</ref> |Mental Health |Peers |Youth |mh_y_peq |abcd_peq01; abcd_mhy02 | | | | |- |Youth Resilience Scale<ref>Ungar, M., & Liebenberg, L. (2009). Cross-cultural consultation leading to the development of a valid measure of youth resilience: The International Resilience Project. ''Studia psychologica'', ''51''(2-3), 259-268.</ref> |Mental Health |Peers |Youth |mh_y_or |abcd_ysr01 | | | | |- |Prodromal Psychosis Scale<ref>Karcher NR, Barch DM, Avenevoli S, Savill M, Huber RS, Simon TJ, Leckliter IN, Sher KJ, Loewy RL. Assessment of the Prodromal Questionnaire-Brief Child Version for Measurement of Self-reported Psychotic like Experiences in Childhood. JAMA Psychiatry. 2018 Aug 1;75(8):853-861.</ref><ref>Loewy, R.L., Bearden, C.E., et al., 2005. The prodromal questionnaire (PQ): preliminary validation of a self-report screening measure for prodromal and psychotic syndromes. Schizophr. Res. 79 (1), 117–125.</ref><ref>Ising, H.K., Veling, W., et al., 2012. The validity of the 16-item version of the Prodromal Questionnaire (PQ-16) to screen for ultra high risk of developing psychosis in the general help-seeking population. Schizophr. Bull. 38 (6), 1288–1296.</ref><ref>Therman, S., Lindgren, M., et al., 2014. Predicting psychosis and psychiatric hospital care among adolescent psychiatric patients with the Prodromal Questionnaire. Schizophr. Res. 158 (1–3), 7–10.</ref> |Mental Health |Psychosis |Youth |mh_y_pps |abcd_mhy02; pps01 | | | | |- |[[wikipedia:Kiddie_Schedule_for_Affective_Disorders_and_Schizophrenia|Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)]] Background Items Survey - Youth<ref name=":6">KAUFMAN, J., BIRMAHER, B., BRENT, D., RAO, U., FLYNN, C., MORECI, P., WILLIAMSON, D., & RYAN, N. (1997). Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data. ''Journal of the American Academy of Child & Adolescent Psychiatry'', ''36''(7), 980–988. <nowiki>https://doi.org/10.1097/00004583-199707000-00021</nowiki></ref><ref name=":7">Kaufman, J., Birmaher, B., Axelson, D., Perepletchikova, F., Brent, D., & Ryan, N. (2016). ''K-SADS-PL DSM-5''. <nowiki>https://pediatricbipolar.pitt.edu/sites/default/files/assets/Clinical%20tools/KSADS/KSADS_DSM_5_SCREEN_Final.pdf</nowiki> </ref><ref name=":2">Townsend, L, Kobak, K., Kearney, C., Milham, M., Andreotti, C., Escalera, J., Alexander, L., Gill, M.K., Birmaher, B., Sylvester, R., Rice, D., Deep, A., Kaufman, J. (2020).  Development of Three Web-Based Computerized Versions of the Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS-COMP) Child Psychiatric Diagnostic Interview: Preliminary Validity Data. Journal of the American Academy of Child and Adolescent Psychiatry, Feb;59(2):309-325. doi:10.1016/j.jaac. PMID: 31108163.</ref><ref name=":3">Kaufman, J., Kobak, K., Birmaher, B., & de Lacy, N. (2021). KSADS-COMP Perspectives on Child Psychiatric Diagnostic Assessment and Treatment Planning. Journal of the American Academy of Child and Adolescent Psychiatry, ''60''(5), 540–542.</ref> |Mental Health |Psychosocial |Youth |mh_y_ksads_bg |abcd_yksad01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems)<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Youth |mh_y_ksads_slp |sleep_problems01 | | | | |- |Cyberbullying Questionnaire<ref>Stewart RW, Drescher CF, Maack DJ, Ebesutani C, Young J. The Development and Psychometric Investigation of the Cyberbullying Scale. J Interpers Violence. 2014 Aug;29(12):2218-2238. doi: 10.1177/0886260513517552. Epub 2014 Jan 14. PMID: 24424252.</ref> |Mental Health |Social |Youth |mh_y_cbb |abcd_cb01 | | | | |- |Life Events Scale - Youth<ref name=":10">Tiet, Q.Q., Bird, H.R., et al., 2001. Relationship between specific adverse life events and psychiatric disorders. J. Abnorm. Child Psychol. 29 (2), 153–164.</ref><ref name=":15">Grant, K.E., Compas, B.E., et al., 2004. Stressors and child and adolescent psychopathology: measurement issues and prospective effects. J. Clin. Child Adolesc. Psychol. 33 (2), 412–425.</ref> |Mental Health |Stress |Youth |mh_y_le |abcd_mhy02; abcd_yle01 | | | | |- |[[Behavioral Inhibition and Behavioral Activation System (BIS/BAS) Scales|PhenX Behavioral Inhibition/Behavioral Approach System (BIS/BAS) Scales]]<ref>Carver, C. & White, T. (1994). Behavioral Inhibition, Behavioral Activation, and affective response to impending reward and punishment: The BIS/BAS Scales. ''Journal of Personality and Social Psychology'', 67(2), 319-333.</ref><ref>Pagliaccio D, Luking KR, Anokhin AP, Gotlib IH, Hayden EP, Olino TM, Peng CZ, Hajcak G, Barch DM. Revising the BIS/BAS Scale to study development: Measurement invariance and normative effects of age and sex from childhood through adulthood. Psychol Assess. 2016 Apr;28(4):429-42. doi: 10.1037/pas0000186. Epub 2015 Aug 24. PMID: 26302106; PMCID: PMC4766059.</ref> |Mental Health |Temperament/Personality |Youth |mh_y_bisbas |abcd_bisbas01; abcd_mhy02 | | | | |- |Urgency, Premeditation, Perseverance, Sensation Seeking, Positive Urgency, Impulsive Behavior (UPPS-P) for Children - Short Form (ABCD Version)<ref>Whiteside, S. P., & Lynam, D. R. (2001). The Five Factor Model and impulsivity: using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669-689. doi: 10.1016/S0191-8869(00)00064-7</ref><ref>Cyders, M. A., Smith, G. T., Spillane, N. S., Fischer, S., Annus, A. M., & Peterson, C. (2007). Integration of impulsivity and positive mood to predict risky behavior: Development and validation of a measure of positive urgency. ''Psychological Assessment, 19''(1), 107–118. <nowiki>https://doi.org/10.1037/1040-3590.19.1.107</nowiki></ref> |Mental Health |Temperament/Personality |Youth |mh_y_upps |abcd_mhy02; abcd_upps01 | | | | |- | | | | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Agoraphobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_ago |agoraphobia_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_gad |generaled_anx_disorder_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Obsessive Compulsive Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_ocd |obs_compulsive_disorder_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Panic Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_pd |panic_disorder_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Separation Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_sep |separation_anxiety_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_sad |social_anxiety_disorder_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Specific Phobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent |mh_p_ksads_phb |specific_phobia_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Autism Spectrum Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Autism Spectrum |Parent |mh_p_ksads_asd |autism_spectrum_dis_p01 | | | | |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|Adult Self Report Survey]]<ref name=":11">Achenbach, T. M., & Rescorla, L. A. (2003). Manual for the ASEBA adult forms & profiles. Research Center for Children, Youth, & Families, University of Vermont, Burlington, VT, USA.</ref> |Mental Health |Broad Psychopathology |Parent |mh_p_asr |pasr01; abcd_asrs01 | | | | |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Behavior Checklist]]<ref name=":11" /> |Mental Health |Broad Psychopathology |Parent |mh_p_abcl |abcd_abcls01; abcd_adbc01 | | | | |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Self-Report]] (psychopathology)<ref name=":11" /> |Mental Health |Broad Psychopathology |Parent | | | | | | |- |[[wikipedia:Child_Behavior_Checklist|Child Behavior Checklist]]<ref>Achenbach TM, Rescorla LA. ''Manual for the ASEBA school-age forms & profiles: an integrated system of mult-informant assessment.'' Burlington: University of Vermont, Research Center for Children, Youth & Families; 2001.</ref> |Mental Health |Broad Psychopathology |Parent |mh_p_cbcl |abcd_cbcl01; abcd_cbcls01 | | | | |- |Family History Assessment Survey<ref>Brown SA, Brumback T, Tomlinson K, Cummins K, Thompson WK, Nagel BJ, De Bellis MD, Hooper SR, Clark DB, Chung T, Hasler BP, Colrain IM, Baker FC, Prouty D, Pfefferbaum A, Sullivan EV, Pohl KM, Rohlfing T, Nichols BN, Chu W, Tapert SF. The National Consortium on Alcohol and NeuroDevelopment in Adolescence (NCANDA): A Multisite Study of Adolescent Development and Substance Use. J Stud Alcohol Drugs. 2015 Nov;76(6):895-908. doi: 10.15288/jsad.2015.76.895. PMID: 26562597; PMCID: PMC4712659.</ref><ref>Rice JP, Reich T, Bucholz KK, Neuman RJ, Fishman R, Rochberg N, Hesselbrock VM, Nurnberger JI Jr, Schuckit MA, Begleiter H. Comparison of direct interview and family history diagnoses of alcohol dependence. Alcohol Clin Exp Res. 1995 Aug;19(4):1018-23. doi: 10.1111/j.1530-0277.1995.tb00983.x. PMID: 7485811.</ref> |Mental Health |Broad Psychopathology |Parent |mh_p_fhx |fhxp201; abcd_fhxssp01; fhxp102 | | | | |- |KSADS Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Parent |mh_p_ksads_ss |abcd_ksad01; ksads2daic_use_only_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Parent |mh_p_ksads_ed |eating_disorders_p01 | | | | |- |Difficulty in Emotion Regulation Scale<ref>Bardeen, J. R., Fergus, T. A., Hannan, S. M., & Orcutt, H. K. (2016). Addressing psychometric limitations of the Difficulties in Emotion Regulation Scale through item modification. Journal of Personality Assessment.</ref><ref>Bunford, N., Dawson, A. E., Evans, S. W., Ray, A. R., Langberg, J. M., Owens, J. S., DuPaul, G. J., & Allan, D. M. (2020). The Difficulties in Emotion Regulation Scale-Parent Report: A Psychometric Investigation Examining Adolescents With and Without ADHD. Assessment, 27(5), 921–940.</ref> |Mental Health |Emotion |Parent |mh_p_ders |diff_emotion_reg_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (ADHD) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |mh_p_ksads_adhd |attn_deficit_hyperactiv_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |mh_p_ksads_cd |abcd_pksadscd01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Oppositional Defiant Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent |mh_p_ksads_odd |opp_defiant_disorder_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Homicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent |mh_p_ksads_hi |homicidality_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent |mh_p_ksads_si |suicidality_p01 | | | | |- |[[OToPS/Measures/7 Up 7 Down Inventory|General Behavior Inventory - Mania]]<ref name=":9" /> |Mental Health |Mood |Parent |mh_p_gbi |abcd_pgbi01; abcd_mhp02 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Parent <ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |mh_p_ksads_bp |bipolar_disorders_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |mh_p_ksads_dep |depressive_disorders_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent |mh_p_ksads_dmdd |disruptive_mood_dysreg_p01 | | | | |- |KSADS Diagnostic Interview for DSM-5 (Psychotic Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosis |Parent |mh_p_ksads_psy |psychosis_p01 | | | | |- |KSADS Background Items Survey - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosocial |Parent |mh_p_ksads_bg |abcd_lpksad01; dibf01; | | | | |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Parent |mh_p_ksads_slp |sleep_problems_p01 | | | | |- |Short Social Responsiveness Scale<ref>Aldridge, F. J., Gibbs, V. M., Schmidhofer, K., & Williams, M. (2012). Investigating the clinical usefulness of the Social Responsiveness Scale (SRS) in a tertiary level, autism spectrum disorder specific assessment clinic. ''Journal of autism and developmental disorders'', ''42''(2), 294–300.</ref><ref>Constantino J. N. (2011). The quantitative nature of autistic social impairment. ''Pediatric research'', ''69''(5 Pt 2), 55R–62R.</ref><ref>Constantino, J. N., Przybeck, T., Friesen, D., & Todd, R. D. (2000). Reciprocal social behavior in children with and without pervasive developmental disorders. ''Journal of developmental and behavioral pediatrics : JDBP'', ''21''(1), 2–11.</ref><ref>Constantino, J. N., & Todd, R. D. (2000). Genetic structure of reciprocal social behavior. ''The American journal of psychiatry'', ''157''(12), 2043–2045.</ref><ref>Constantino, J. N., & Todd, R. D. (2003). Autistic traits in the general population: a twin study. ''Archives of general psychiatry'', ''60''(5), 524–530.</ref><ref>Constantino, J. N., Gruber, C. P., Davis, S., Hayes, S., Passanante, N., & Przybeck, T. (2004). The factor structure of autistic traits. ''Journal of child psychology and psychiatry, and allied disciplines'', ''45''(4), 719–726.</ref><ref>Hus, V., Bishop, S., Gotham, K., Huerta, M., & Lord, C. (2013). Factors influencing scores on the social responsiveness scale. ''Journal of child psychology and psychiatry, and allied disciplines'', ''54''(2), 216–224.</ref><ref>Kaat, A. J., & Farmer, C. (2017). Commentary: Lingering questions about the Social Responsiveness Scale short form. A commentary on Sturm et al. (2017). ''Journal of child psychology and psychiatry, and allied disciplines'', ''58''(9), 1062–1064.</ref><ref>Norris, M., & Lecavalier, L. (2010). Screening accuracy of Level 2 autism spectrum disorder rating scales. A review of selected instruments. ''Autism : the international journal of research and practice'', ''14''(4), 263–284.</ref> |Mental Health |Social |Parent |mh_p_ssrs |abcd_mhp02; abcd_pssrs01; | | | | |- |KSADS Diagnostic Interview for DSM-5 (Post-Traumatic Stress Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Stress |Parent |mh_p_ksads_ptsd |abcd_ptsd01 | | | | |- |Life Events Scale - Parent<ref name=":10" /><ref name=":15" /> |Mental Health |Stress |Parent |mh_p_le |abcd_mhp02; abcd_ple01 | | | | |- |[[wikipedia:Perceived_Stress_Scale|Perceived Stress Scale]]<ref>Cohen, S., Kamarck, T., and Mermelstein, R. (1983). A global measure of perceived stress. Journal of Health and Social Behavior, 24, 386-396. </ref> |Mental Health |Stress |Parent |mh_p_pss |abcd_mhy02; perceived_stress_p01 | | | | |- |Early Adolescent Temperament Questionnaire<ref>Latham MD, Dudgeon P, Yap MBH, Simmons JG, Byrne ML, Schwartz OS, Ivie E, Whittle S, Allen NB. Factor Structure of the Early Adolescent Temperament Questionnaire-Revised. Assessment. 2020 Oct;27(7):1547-1561. doi: 10.1177/1073191119831789. Epub 2019 Feb 21. PMID: 30788984.</ref> |Mental Health |Temperament/Personality |Parent |mh_p_eatq |abcd_mhp02; abcd_eatqp01 | | | | |- | | | | | | | | | | |- |Brief Problem Monitor |Mental Health |Broad Psychopathology |Teacher |mh_t_bpm |abcd_ssbpmtf01; abcd_bpmt01 | | | | |- | | | | | | | | | | |- |[[wikipedia:Edinburgh_Handedness_Inventory|Edinburgh Handedness Inventory]] |Neurocognition |Administrative |Youth |nc_y_eihs |abcd_ehis01 | | | | |- |Neurocognition Assessment Administration |Neurocognition |Administrative |Youth |nc_y_adm |neurocog_youth_session01 | | | | |- |[[wikipedia:Snellen_chart|Snellen Vision Screener]] |Neurocognition |Administrative |Youth |nc_y_svs |abcd_svs01 | | | | |- |Behavioral Indicator of Resiliency to Distress Task (BIRD)<ref>Lejuez, C. W., Kahler, C. W., & Brown, R. A. (2003). 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''Behavior genetics'', ''41''(2), 175–183.</ref> |Neurocognition |Tasks |Youth |nc_y_cct |cct01 | | | | |- |Delay Discounting Task<ref>Johnson, M. W., & Bickel, W. K. (2008). An algorithm for identifying nonsystematic delay-discounting data. ''Experimental and clinical psychopharmacology'', ''16''(3), 264–274.</ref> |Neurocognition |Tasks |Youth |nc_y_ddis |abcd_yddss01 | | | | |- |Emotional Faces Stroop Task<ref>Başgöze, Z., Gönül, A. S., Baskak, B., & Gökçay, D. (2015). Valence-based Word-Face Stroop task reveals differential emotional interference in patients with major depression. ''Psychiatry research'', ''229''(3), 960–967.</ref><ref>Kane, M. J., & Engle, R. W. (2003). Working-memory capacity and the control of attention: the contributions of goal neglect, response competition, and task set to Stroop interference. ''Journal of experimental psychology. General'', ''132''(1), 47–70.</ref><ref>Stroop, J.R., 1935. Studies of interference in serial verbal reactions. J. Exp. Psychol. 18 (6), 643–662.</ref> |Neurocognition |Tasks |Youth |nc_y_est |abcd_yest01 | | | | |- |Game of Dice Task<ref>Brand, M., Fujiwara, E., Borsutzky, S., Kalbe, E., Kessler, J., & Markowitsch, H. J. (2005). Decision-making deficits of Korsakoff patients in a new gambling task with explicit rules: associations with executive functions. ''Neuropsychology'', ''19''(3), 267–277.</ref><ref>Drechsler, R., Rizzo, P., & Steinhausen, H. C. (2008). Decision-making on an explicit risk-taking task in preadolescents with attention-deficit/hyperactivity disorder. ''Journal of neural transmission (Vienna, Austria : 1996)'', ''115''(2), 201–209.</ref><ref>Duperrouzel, J. C., Hawes, S. W., Lopez-Quintero, C., Pacheco-Colón, I., Coxe, S., Hayes, T., & Gonzalez, R. (2019). Adolescent cannabis use and its associations with decision-making and episodic memory: Preliminary results from a longitudinal study. ''Neuropsychology'', ''33''(5), 701–710.</ref><ref>Ross, J. M., Graziano, P., Pacheco-Colón, I., Coxe, S., & Gonzalez, R. (2016). Decision-Making Does not Moderate the Association between Cannabis Use and Body Mass Index among Adolescent Cannabis Users. ''Journal of the International Neuropsychological Society : JINS'', ''22''(9), 944–949.</ref> |Neurocognition |Tasks |Youth |nc_y_gdt |abcd_gdss01; neurocog_youth_session01 | | | | |- |Little Man Task<ref>Acker, W. (1982). “A computerized approach to psychological screening—The Bexley-Maudsley Automated Psychological Screening and The Bexley-Maudsley Category Sorting Test.” ''International Journal of Man-Machine Studies'', ''17''(3): 361-369.</ref><ref>Nixon, S. J., Prather, R. A., & Lewis, B. (2014). Sex differences in alcohol-related neurobehavioral consequences. In Edith V. Sullivan and Adolf Pfefferbaum (Eds.), Alcohol and the nervous system (Handbook of clinical neurology, 3rd series (Vol. 125)). Oxford, United Kingdom, Elsevier, pp. 253-272.</ref> |Neurocognition |Tasks |Youth |nc_y_lmt |neurocog_youth_session01; lmtp201 | | | | |- |NIH Toolbox Tasks - Dimensional Change Card Sort<ref>{{Cite web|url=https://nihtoolbox.org/test/dimensional-change-card-sort-test/|title=Dimensional Change Card Sort Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |nc_y_nihtb |abcd_tbss01; neurocog_youth_session01 | | | | |- |NIH Toolbox Tasks - Flanker Inhibitory Control & Attention<ref>{{Cite web|url=https://nihtoolbox.org/test/flanker-inhibitory-control-and-attention-test-age-12/|title=Flanker Inhibitory Control and Attention Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |nc_y_flkr; nc_y_nihtb |abcd_tbss01; neurocog_youth_session01 | | | | |- |NIH Toolbox Tasks - Oral Reading Recognition<ref>{{Cite web|url=https://nihtoolbox.org/test/a-dummy-iq-test/|title=Oral Reading Recognition Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |nc_y_nihtb |abcd_tbss01; neurocog_youth_session01 | | | | |- |NIH Toolbox Tasks - Pattern Comparison Processing Speed<ref>{{Cite web|url=https://nihtoolbox.org/test/pattern-comparison-processing-speed/|title=Pattern Comparison Processing Speed Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |nc_y_nihtb |abcd_tbss01; neurocog_youth_session01 | | | | |- |NIH Toolbox Tasks - Picture Sequence Memory<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-sequence-memory-test/|title=Picture Sequence Memory Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth |nc_y_nihtb |abcd_tbss01; neurocog_youth_session01 | | | | |- |NIH Toolbox Tasks - Picture Vocabulary<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-vocabulary-test/|title=Picture Vocabulary Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12">McDonald, Skye (Ed.) 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Oxford University Press. New York, NY.</ref> |Neurocognition |Tasks |Youth |nc_y_ravlt |neurocog_youth_session01; abcd_ps01 | | | | |- |Social Influence Task<ref>Knoll, L. J., Leung, J. T., Foulkes, L., & Blakemore, S. J. (2017). Age-related differences in social influence on risk perception depend on the direction of influence. ''Journal of adolescence'', ''60'', 53–63.</ref> |Neurocognition |Tasks |Youth |nc_y_sit |neurocog_youth_session01; abcd_siss01 | | | | |- |Stanford Mental Arithmetic Response Time Evaluation (SMARTE)<ref>Starkey, G. S., & McCandliss, B. D. (2014). The emergence of “groupitizing” in children’s numerical cognition. ''Journal of experimental child psychology'', ''126'', 120–137.</ref> |Neurocognition |Tasks |Youth |nc_y_smarte |smarte_sumscores01 | | | | |- |Wechsler Intelligence Scale for Children - Matrix Reasoning Task<ref>Wechsler, D. (2014). Wechsler Intelligence Scale for Children - Fifth Edition Manual. 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''Barkley Deficits in Executive Functioning Scale--Children and Adolescents (BDEFS-CA)'': Guilford Press.</ref> |Neurocognition |Questionnaire |Parent |nc_p_bdef |barkley_exec_func01 | | | | |- | | | | | | | | | | |- |[[wikipedia:Breathalyzer|Alcohol Toxicology]] |Substance Use |Drug Toxicology |Youth |su_y_alc_tox |yalcs01 | | | | |- |Hair Drug Toxicology<ref name=":41">Wade, N. E., Tapert, S. F., Lisdahl, K. M., Huestis, M. A., & Haist, F. (2022). Substance use onset in high-risk 9-13 year-olds in the ABCD study. Neurotoxicol Teratol, 91, 107090.</ref><ref name=":42">Wade, N. E., Sullivan, R. M., Tapert, S. F., Pelham, W. E., 3rd, Huestis, M. A., Lisdahl, K. M., & Haist, F. (2023). Concordance between substance use self-report and hair analysis in community-based adolescents. Am J Drug Alcohol Abuse, 49(1), 76-84.</ref> |Substance Use |Drug Toxicology |Youth |su_y_hair_tox |N/A | | | | |- |Nicotine Toxicology<ref name=":4">''NicAlert | NYMOX''. (2024). Nymox.com. <nowiki>https://nymox.com/products/nicalert</nowiki> </ref><ref name=":38">''Alere iScreen Dip Card''. (2024). Globalpointofcare.abbott. <nowiki>https://www.globalpointofcare.abbott/us/en/product-details/toxicology-iscreen.html</nowiki></ref><ref name=":39">''Reditest® Smoke Cassette | Test Devices | Redwood Toxicology Laboratory''. (2024). Redwoodtoxicology.com. <nowiki>https://www.redwoodtoxicology.com/devices/doa_redismoke</nowiki></ref><ref name=":40">''ReditestTM Smoke Cassette''. (2024). Toxicology.abbott. <nowiki>https://www.toxicology.abbott/us/en/products/reditest-smoke-cassette.html</nowiki></ref> |Substance Use |Drug Toxicology |Youth |su_y_nic_tox |abcd_yn01 | | | | |- |Saliva Drug Toxicology |Substance Use |Drug Toxicology |Youth |su_y_sal_tox |abcd_ytt01 | | | | |- |Urine Drug Toxicology<ref name=":38" /> |Substance Use |Drug Toxicology |Youth |su_y_uri_tox |N/A | | | | |- |Alcohol Expectancies Questionnaire (AEQ-AB)<ref>Brown, S. A., Christiansen, B. A., & Goldman, M. S. (1987). The Alcohol Expectancy Questionnaire: an instrument for the assessment of adolescent and adult alcohol expectancies. ''Journal of studies on alcohol'', ''48''(5), 483–491.</ref><ref>Greenbaum, P. E., Brown, E. C., & Friedman, R. M. (1995). Alcohol expectancies among adolescents with conduct disorder: prediction and mediation of drinking. ''Addictive behaviors'', ''20''(3), 321–333.</ref><ref>Stein, L. A., Katz, B., Colby, S. M., Barnett, N. P., Golembeske, C., Lebeau-Craven, R., & Monti, P. M. (2007). Validity and Reliability of the Alcohol Expectancy Questionnaire-Adolescent, Brief. ''Journal of child & adolescent substance abuse'', ''16''(2), 115–127.</ref> |Substance Use |SU Attitude |Youth |su_y_alc_exp |abcd_yam01; abcd_suss01 | | | | |- |Alcohol Motives Questionnaire (PhenX)<ref>Cooper, M. L. (1994). Motivations for alcohol use among adolescents: Development and validation of a four-factor model. Psychological Assessment, 6, 117−128.</ref><ref>Grant, V. V., Stewart, S. H., O’Connor, R. M., Blackwell, E., & Conrod, P. J. (2007). Psychometric evaluation of the five-factor Modified Drinking Motives Questionnaire–Revised in undergraduates. ''Addictive behaviors'', ''32''(11), 2611–2632.</ref><ref>Kuntsche, E., & Kuntsche, S. (2009). Development and validation of the Drinking Motive Questionnaire Revised Short Form (DMQ-R SF). Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53, ''38''(6), 899–908.</ref> |Substance Use |SU Attitude |Youth |su_y_alc_motiv |abcd_yam01 | | | | |- |Cigarette Expectancies (ASCQ)<ref>Lewis-Esquerre, J. M., Rodrigue, J. R., & Kahler, C. W. (2005). Development and validation of an adolescent smoking Consequence questionnaire. ''Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco'', ''7''(1), 81–90.</ref> |Substance Use |SU Attitude |Youth |su_y_nic_exp |abcd_suss01; abcd_ynm01 | | | | |- |Electronic Nictotine Delivery Systems Expectancies Questionnaire<ref>Pokhrel, P., Lam, T.H., Pagano, I., Kawamoto, C.T., & Herzog, T.A. (2018). YPokhrel, P., Lam, T. H., Pagano, I., Kawamoto, C. T., & Herzog, T. A. (2018). Young adult e-cigarette use outcome expectancies: Validity of a revised scale and a short scale. ''Addictive behaviors'', ''78'', 193–199.</ref> |Substance Use |SU Attitude |Youth |su_y_ends_exp |abcd_suss01; abcd_ynm01 | | | | |- |Electronic Nicotine Delivery System Motives Inventory<ref>Centers for Disease Control (CDC; Division of Nutrition). (2016). Anthropometry Procedures Manual.</ref><ref name=":24">Piper, M. E., Piasecki, T. M., Federman, E. B., Bolt, D. M., Smith, S. S., Fiore, M. C., & Baker, T. B. (2004). A multiple motives approach to tobacco dependence: the Wisconsin Inventory of Smoking Dependence Motives (WISDM-68). Journal of consulting and clinical psychology, ''72''(2), 139–154.</ref> |Substance Use |SU Attitude |Youth |su_y_ends_motiv |abcd_ynm01 | | | | |- |Marijuana Effect Expectancy Questionnaire (MEEQ-B)<ref name=":21">Torrealday, O., Stein, L. A., Barnett, N., Golembeske, C., Lebeau, R., Colby, S. M., & Monti, P. M. (2008). Validation of the Marijuana Effect Expectancy Questionnaire-Brief. ''Journal of child & adolescent substance abuse'', ''17''(4), 1–17.</ref> |Substance Use |SU Attitude |Youth |su_y_can_exp |abcd_suss01; abcd_ymidm01 | | | | |- |Marijuana Motives Questionnaire (PhenX)<ref>Lee, C. M., Neighbors, C., Hendershot, C. S., & Grossbard, J. R. (2009). Development and preliminary validation of a comprehensive marijuana motives questionnaire. Journal of studies on alcohol and drugs, ''70''(2), 279–287.</ref><ref>Simons, J., Correia, C. J., Carey, K. 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Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, ''17''(7), 862–869.</ref> |Substance Use |SU Attitude |Youth |su_y_path_intuse |abcd_ysu02; abcd_ysua01 | | | | |- |PhenX Peer Group Deviance Survey<ref>Freedman, D., Thornton, A., Camburn, D., Alwin, D., & Young-demarco, L. (1988). The life history calendar: a technique for collecting retrospective data. Sociological methodology, ''18'', 37–68.</ref> |Substance Use |SU Attitude |Youth |su_y_peerdevia |abcd_ysua01; abcd_ysu02 | | | | |- |PhenX Peer Tolerance of Use<ref name=":19" /><ref name=":20" /> |Substance Use |SU Attitude |Youth |su_y_peertoler |abcd_ysua01 | | | | |- |PhenX Perceived Harm of Substance Use<ref name=":19">Johnston, Lloyd D.; O’Malley, P. M.; Bachman, J. G.; Schulenberg, J. E.. (2009). Monitoring the Future. National Results on Adolescent Drug Use: Overview of Key Findings, 2009. NIH Publication Number 10-7583</ref><ref name=":20">Miech, R. A.; Johnston, L. 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''Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco'', ''12''(5), 489–499.</ref> |Substance Use |SU Attitude |Youth |su_y_nic_motiv |abcd_ynm01 | | | | |- |Vaping Expectancies<ref name=":21" /> |Substance Use |SU Attitude |Youth |su_y_vap_exp |abcd_ymidm01 | | | | |- |Vaping Motives<ref>Diez, S. L., Cristello, J. V., Dillon, F. R., De La Rosa, M., & Trucco, E. M. (2019). Validation of the electronic cigarette attitudes survey (ECAS) for youth. Addictive behaviors, ''91'', 216–221.</ref> |Substance Use |SU Attitude |Youth |su_y_vap_motiv |abcd_ymidm01 | | | | |- |Alcohol Hangover Symptoms (HSS)<ref>Slutske, W. S., Piasecki, T. M., & Hunt-Carter, E. E. (2003). Development and initial validation of the Hangover Symptoms Scale: prevalence and correlates of Hangover Symptoms in college students. 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Marijuana use motives: A confirmatory test and evaluation among young adult marijuana users. Addictive behaviors, ''32''(12), 3122–3130.</ref> |Substance Use |SU Consequence |Youth |su_y_can_mapi |abcd_ymidm01; abcd_ysu02 | | | | |- |Marijuana Subjective Effects<ref>Agrawal, A., Madden, P. A., Bucholz, K. K., Heath, A. C., & Lynskey, M. T. (2014). Initial reactions to tobacco and cannabis smoking: a twin study. Addiction (Abingdon, England), ''109''(4), 663–671.</ref><ref>Agrawal, A., Madden, P. A., Martin, N. G., & Lynskey, M. T. (2013). Do early experiences with cannabis vary in cigarette smokers?. ''Drug and alcohol dependence'', ''128''(3), 255–259.</ref> |Substance Use |SU Consequence |Youth |su_y_can_eff |abcd_suss01; abcd_ymidm01; abcd_ysu02 | | | | |- |Nicotine Dependence (PATH)<ref name=":22">Pomerleau, O. F., Pomerleau, C. S., & Namenek, R. J. (1998). Early experiences with tobacco among women smokers, ex-smokers, and never-smokers. Addiction (Abingdon, England), ''93''(4), 595–599.</ref><ref>Prokhorov, A. V., Pallonen, U. E., Fava, J. L., Ding, L., & Niaura, R. (1996). Measuring nicotine dependence among high-risk adolescent smokers. Addictive behaviors, ''21''(1), 117–127.</ref> |Substance Use |SU Consequence |Youth |su_y_nic_dep |abcd_ynm01 | | | | |- |Nicotine Subjective Response<ref name=":22" /><ref>Rodriguez, D., & Audrain-McGovern, J. (2004). Construct validity analysis of the early smoking experience questionnaire for adolescents. Addictive behaviors, ''29''(5), 1053–1057.</ref> |Substance Use |SU Consequence |Youth |su_y_nic_eff |abcd_suss01; abcd_ynm01; abcd_ysu02 | | | | |- |[[Evidence-based assessment/Rx4DxTx of SubstanceUse|Rutgers Alcohol Problem Index (RAPI)]]<ref>White, H. R., & Labouvie, E. W. (1989). Towards the assessment of adolescent problem drinking. Journal of studies on alcohol, ''50''(1), 30–37.</ref> |Substance Use |SU Consequence |Youth |su_y_alc_rapi |abcd_yam01; abcd_ysu02; | | | | |- |PhenX Community Risk and Protective Factors<ref name=":25">Arthur, M. W., Briney, J. S., Hawkins, J. D., Abbott, R. D., Brooke-Weiss, B. L., & Catalano, R. F. (2007). Measuring risk and protection in communities using the Communities That Care Youth Survey. Evaluation and program planning, ''30''(2), 197–211.</ref><ref name=":26">Trentacosta, C. J., Criss, M. M., Shaw, D. S., Lacourse, E., Hyde, L. W., & Dishion, T. J. (2011). Antecedents and outcomes of joint trajectories of mother-son conflict and warmth during middle childhood and adolescence. Child development, ''82''(5), 1676–1690.</ref> |Substance Use |SU Environment |Youth |su_y_crpf |abcd_ycrpf01 | | | | |- |Sibling Use<ref>Samek, D. R., Goodman, R. J., Riley, L., McGue, M., & Iacono, W. G. (2018). The Developmental Unfolding of Sibling Influences on Alcohol Use over Time. Journal of youth and adolescence, ''47''(2), 349–368.</ref> |Substance Use |SU Environment |Youth |su_y_sibuse |abcd_ysua01 | | | | |- |Substance Use Interview <ref name=":17" /><ref name=":18" /> |Substance Use |Substance Use |Youth |su_y_sui |abcd_suss01; abcd_ysu02; abcd_ysuip01 |Caffeine Intake Survey; ISay II Q2 Sipping Items, Low Level Marijuana Use, Low Level Tobacco Use | | | |- |Participant Last Use Survey (PLUS) (Day 1/2/3/4) - Youth<ref name=":17">Lisdahl, K. M., Sher, K. J., Conway, K. P., Gonzalez, R., Feldstein Ewing, S. W., Nixon, S. J., Tapert, S., Bartsch, H., Goldstein, R. Z., & Heitzeg, M. (2018). Adolescent brain cognitive development (ABCD) study: Overview of substance use assessment methods. Developmental cognitive neuroscience, ''32'', 80–96.</ref> |Substance Use |Substance Use |Youth |su_y_plus |abcd_plus01 | | | | |- |Substance Use Phone Interview<ref name=":17" /> |Substance Use |Substance Use |Youth |su_y_mypi |abcd_ymypisu01 | | | | |- |Timeline Follow-Back Survey<ref name=":17" /><ref>Sobell, L. C., & Sobell, M. B. (1996). Time Line Follow Back. User s Guide, Toronto. ''Addiction Research Foundation''.</ref> |Substance Use |Substance Use |Youth |su_y_tlfb |abcd_tlfb01 | | | | |- |Opportunity to Use Questionnaire |Substance Use | |Youth | | | | | | |- | | | | | | | | | | |- |KSADS - Alcohol/Drug Use Disorder - Parent<ref name=":3" /> |Substance Use |SU Consequence |Parent |su_p_ksads_sud |alcohol_use_disorder_p01; drug_use_disorders_p01 | | | | |- |Household Substance Use, Density, Storage & Second-Hand Exposure<ref>Bartels, K., Mayes, L. M., Dingmann, C., Bullard, K. J., Hopfer, C. J., & Binswanger, I. A. (2016). Opioid Use and Storage Patterns by Patients after Hospital Discharge following Surgery. ''PloS one'', ''11''(1), e0147972.</ref><ref>Friese, B., Grube, J. W., & Moore, R. S. (2012). How parents of adolescents store and monitor alcohol in the home. ''The journal of primary prevention'', ''33''(2-3), 79–83.</ref> |Substance Use |SU Environment |Parent |su_p_dse |abcd_pssudse01 | | | | |- |Parent Rules Survey<ref>Dishion, T.J., Kavanagh, K., 2003. Intervening in Adolescent Problem Behavior: A Family-centered Approach. The Guilford Press, New York, NY.</ref><ref>Dishion, T. J., Nelson, S. E., & Kavanagh, K. (2003). The family check-up with high-risk young adolescents: Preventing early-onset substance use by parent monitoring. Behavior Therapy, ''34''(4), 553-571.</ref><ref name=":18" /><ref>Jackson, K. M., Roberts, M. E., Colby, S. M., Barnett, N. P., Abar, C. C., & Merrill, J. E. (2014). Willingness to drink as a function of peer offers and peer norms in early adolescence. ''Journal of studies on alcohol and drugs'', ''75''(3), 404–414.</ref> |Substance Use |SU Environment |Parent |su_p_pr |prq01 | | | | |- |PhenX Community Risk and Protective Factors<ref name=":25" /><ref name=":26" /> |Substance Use |SU Environment |Parent |su_p_crpf |abcd_crpf01 | | | | |- |Participant Last Use Survey (PLUS) (Day 1/2/3/4) - Parent<ref name=":17" /> |Substance Use |Substance Use |Parent |su_p_plus |plus01 | | | | |- | | | | | | | | | | |- |Wills Problem Solving Scale<ref>Wills, T. A., Ainette, M. G., Stoolmiller, M., Gibbons, F. X., & Shinar, O. (2008). Good self-control as a buffering agent for adolescent substance use: an investigation in early adolescence with time-varying covariates. ''Psychology of addictive behaviors : journal of the Society of Psychologists in Addictive Behaviors'', ''22''(4), 459–471.</ref> |Culture & Environment |Cognition |Youth |ce_y_wps |abcd_ywpss01; abcd_sscey01 | | | | |- |Perceived Discrimination Scale<ref>Garnett, B. R., Masyn, K. E., Austin, S. B., Miller, M., Williams, D. R., & Viswanath, K. (2014). The intersectionality of discrimination attributes and bullying among youth: an applied latent class analysis. Journal of youth and adolescence, 43(8), 1225–1239.</ref><ref>Phinney, J. S., Madden, T., & Santos, L. J. (1998). Psychological variables as predictors of perceived ethnic discrimination among minority and immigrant adolescents. ''Journal of Applied Social Psychology, 28''(11), 937–953</ref> |Culture & Environment |Community |Youth |ce_y_dm |abcd_ydmes01, abcd_sscey01 | | | | |- |PhenX Neighborhood Safety/Crime Survey - Youth<ref name=":33">Mujahid, M. S., Diez Roux, A. V., Morenoff, J. D., & Raghunathan, T. (2007). Assessing the measurement properties of neighborhood scales: from psychometrics to ecometrics. ''American journal of epidemiology'', ''165''(8), 858–867.</ref> |Culture & Environment |Community |Youth |ce_y_nsc |abcd_nsc01 | | | | |- |Mexican American Cultural Values Scale - Youth<ref>Knight, G.P., Gonzales, N.A., Saenz, D.S., Bonds, D.D., German, M., Deardorff, J., Roosav, M.W., Updegraff, K.A., 2010. The Mexican American cultural values scale for adolescents and adults. J. Early Adolesc. 30 (3), 444–481.</ref> |Culture & Environment |Culture |Youth |ce_y_macv |abcd_macvsy01; abcd_sscey01 | | | | |- |Multi-Group Ethnic Identity - Revised - Youth<ref name=":36">Phinney, J. S., & Ong, A. D. (2007). Conceptualization and measurement of ethnic identity: Current status and future directions. Journal of Counseling Psychology, ''54''(3), 271-281.</ref> |Culture & Environment |Culture |Youth |ce_y_meim |multigrp_ethnic_id_meim01 | | | | |- |Native American Acculturation Survey - Youth<ref name=":37">Garrett MT, Pichette EF. Red as an apple: Native American acculturation and counseling with or without reservation. Journal of Counseling and Development. 2000;78:3–13. </ref> |Culture & Environment |Culture |Youth | | | | | | |- |PhenX Acculturation Survey - Youth<ref name=":28">Alegria, M., Takeuchi, D., Canino, G., Duan, N., Shrout, P., Meng, X.-L., Gong, F., et al. (2004). Considering context, place, and culture: the national Latino and Asian American study. Int. J. Methods Psychiatr. Res. 13 (4), 208–22.</ref><ref name=":29">Marin, G., F. Sabogal, B. V. Marin, R. Otero-Sabogal and E. J. Perez-Stable (1987). “Development of a Short Acculturation Scale for Hispanics.” Hispanic Journal of Behavioral Sciences 9(2): 183-205.</ref> |Culture & Environment |Culture |Youth |ce_y_acc |yacc01 | | | | |- |Vancouver Index of Acculturation - Youth<ref name=":35">Ryder, A.G., Alden, L.E., Paulhus, D.L., 2000. Is acculturation unidimensional or bidimensional? A head-to-head comparison in the prediction of personality, self-identity, and adjustment. J. Pers. Soc. Psychol. 79 (1), 49–65.</ref> |Culture & Environment |Culture |Youth |ce_y_via |vancouver_identity_accult01 | | | | |- |Pet Ownership<ref>Purweal, R., Christley, R., Kordas, K., Joinson, C., Meints, K., Gee, N., & Westgarth, C. (2017). Companion animals and child/adolescent development: A systematic review of the evidence. International Journal of Environmental Research and Public Health, 14(3), 234-259.</ref> |Culture & Environment |Family |Youth |ce_y_pet |pet_ownership01 | | | | |- |PhenX [[wikipedia:Family_Environment_Scale|Family Environment Scale]] - Family Conflict - Youth<ref name=":32">Moos, R.H., Moos, B.S. (1994). Family Environment Scale Manual. Consulting Psychologists Press, Palo Alto, CA.</ref> |Culture & Environment |Family |Youth |ce_y_fes |abcd_fes01; abcd_sscey01 | | | | |- |Acceptance Subscale from Children's Report of Parental Behavior Inventory (CRPBI) - Short<ref>Schaefer, E.S., 1965. A configurational analysis of children’s reports of parent behavior. J. Consult. Psychol. 29, 552–557.</ref><ref>Schludermann, E. H., & Schludermann, S. M. (1988). Children’s Report on Parent Behavior (CRPBI-108, CRPBI-30) for older children and adolescents. Winnipeg, MB, Canada: University of Manitoba.</ref><ref>Barber, B. K., Olsen, J. E., & Shagle, S. C. (1994). Associations between parental psychological and behavioral control and youth internalized and externalized behaviors. Child development, 65(4), 1120-1136.</ref><ref>Barber, B. K., & Olsen, J. A. (1997). Socialization in context: Connection, regulation, and autonomy in the family, school, and neighborhood, and with peers. Journal of Adolescent Research, 12(2), 287-315</ref> |Culture & Environment |Parenting |Youth |ce_y_crpbi |abcd_sscey01; crpbi01 | | | | |- |Multidimensional Neglectful Behavior Scale<ref>Dubowitz, H., Villodas, M. T., Litrownik, A. J., Pitts, S. C., Hussey, J. M., Thompson, R., … & Runyan, D. (2011). Psychometric properties of a youth self-report measure of neglectful behavior by parents. Child Abuse & Neglect, 35(6), 414-424.</ref> |Culture & Environment |Parenting |Youth |ce_y_mnbs |abcd_sscey01; neglectful_behavior01 | | | | |- |Parental Monitoring Survey<ref>Chilcoat, H. D., & Anthony, J. C. (1996). Impact of parent monitoring on initiation of drug use through late childhood. Journal of the American Academy of Child and Adolescent Psychiatry, ''35''(1), 91–100.</ref><ref name=":30">Karoly, H. C., Callahan, T., Schmiege, S. J., & Feldstein Ewing, S. W. (2015). Evaluating the Hispanic Paradox in the context of adolescent risky sexual behavior: the role of parent monitoring. Journal of pediatric psychology, 41(4), 429-440.</ref><ref name=":31">Stattin, H., & Kerr, M. (2000). Parental monitoring: a reinterpretation. Child development, ''71''(4), 1072–1085.</ref> |Culture & Environment |Parenting |Youth |ce_y_pm |abcd_sscey01; pmq01 | | | | |- |Peer Behavior Profile: Prosocial Peer Involvement & Delinquent Peer Involvement<ref>Bingham, C. R., Fitzgerald, H. E., & Zucker, R. A. (1995). Peer Behavior Profile/Peer Activities Questionnaire. Unpublished questionnaire. Department of Psychology, Michigan State University. East Lansing.</ref><ref>Hirschi, T. (1969). Causes of delinquency. Berkeley, CA: University of California Press.</ref><ref>Jessor, R., & Jessor, S.L. (1977). Problem behavior and psychosocial development: A longitudinal study of youth. New York, Academic Press.</ref> |Culture & Environment |Peers |Youth |ce_y_pbp |abcd_pbp01; abcd_sscey01 | | | | |- |Peer Network Health: Protective Scale<ref>Mason, M., Light, J., Campbell, L., Keyser-Marcus, L., Crewe, S., Way, T., Saunders, H., King, L., Zaharakis, N.M., & McHenry, C. (2015). Peer network counseling with urban adolescents: A randomized controlled trial with moderate substance users. Journal of Substance Abuse Treatment, 58, 16-24.</ref> |Culture & Environment |Peers |Youth |ce_y_pnh |abcd_pnhps01; abcd_sscey01 | | | | |- |Resistance to Peer Influence Scale/Questionnaire<ref>Steinberg, L., & Monahan, K. C. (2007). Age differences in resistance to peer influence. ''Developmental psychology'', ''43''(6), 1531–1543.</ref> |Culture & Environment |Peers |Youth |ce_y_rpi | | | | | |- |PhenX School Risk & Protective Factors Survey<ref>Arthur, M. W., Briney, J. S., Hawkins, J. D., Abbott, R. D., Brooke-Weiss, B. L., & Catalano, R. F. (2007). Measuring risk and protection in communities using the Communities That Care Youth Survey. ''Evaluation and program planning'', ''30''(2), 197–211.</ref><ref>Hamilton, C. M., Strader, L. C., Pratt, J. G., Maiese, D., Hendershot, T., Kwok, R. K., Hammond, J. A., Huggins, W., Jackman, D., Pan, H., Nettles, D. S., Beaty, T. H., Farrer, L. A., Kraft, P., Marazita, M. L., Ordovas, J. M., Pato, C. N., Spitz, M. R., Wagener, D., Williams, M., … Haines, J. (2011). The PhenX Toolkit: get the most from your measures. American journal of epidemiology, ''174''(3), 253–260.</ref> |Culture & Environment |School |Youth |ce_y_srpf |srpf01; abcd_sscey01 | | | | |- |School Attendance of Youth & Grades<ref name=":34">Zucker RA, Gonzalez R, Feldstein Ewing SW, Paulus MP, Arroyo J, Fuligni A, Morris AS, Sanchez M, Wills T. Assessment of culture and environment in the Adolescent Brain and Cognitive Development Study: Rationale, description of measures, and early data. Dev Cogn Neurosci. 2018 Aug;32:107-120</ref> |Culture & Environment |School |Youth |ce_y_sag |abcd_ysaag01 | | | | |- |Prosocial Behavior Survey - Youth<ref name=":27">Goodman, R., Meltzer, H., Bailey, V., 1998. The strengths and difficulties questionnaire: a pilot study on the validity of the self-report version. Eur. Child Adolesc. Psychiatry 7(3), 125–130.</ref> |Culture & Environment |Temperament/Personality |Youth |ce_y_psb |abcd_psb01; abcd_sscey01 | | | | |- |Activity Space |Culture & Environment | |Youth | | | | | | |- | | | | | | | | | | |- |Community Cohesion (PhenX)<ref>National Archive of Criminal Justice Data (NACJD), Project on Human Development in Chicago Neighborhoods (PHDCN). Community Survey 1994-1995.</ref><ref>PhenX Protocol - Neighborhood Collective Efficacy - Community Cohesion and Informal Social Control.</ref> |Culture & Environment |Community |Parent |ce_p_comc |abcd_pxccp01; abcd_sscep01 | | | | |- |PhenX Neighborhood Safety/Crime Survey - Parent<ref>Echeverria, S. E., Diez-Roux, A. V., et al. (2004) Reliability of self-reported neighborhood characteristics. J Urban Health 81(4): 682-701.</ref><ref name=":33" /> |Culture & Environment |Community |Parent |ce_p_nsc |abcd_pnsc01; abcd_sscep01 | | | | |- |Mexican American Cultural Values Scale<ref name=":36" /> |Culture & Environment |Culture |Parent |ce_p_macv |abcd_sscep01; macv01 | | | | |- |Multi-Group Ethnic Identity Survey<ref name=":36" /> |Culture & Environment |Culture |Parent |ce_p_meim |abcd_meim01; abcd_sscep01 | | | | |- |Native American Acculturation Scale<ref name=":37" /> |Culture & Environment |Culture |Parent | | | | | | |- |PhenX Acculturation Survey - Parent<ref name=":28" /><ref name=":29" /> |Culture & Environment |Culture |Parent |ce_p_acc |pacc01 | | | | |- |Vancouver Index of Acculturation - Parent<ref name=":35" /> |Culture & Environment |Culture |Parent |ce_p_via |abcd_via01; abcd_sscep01 | | | | |- |PhenX Family Environment Scale - Family Conflict - Parent<ref name=":32" /><ref>Sanford, K., Bingham, C.R., & Zucker, R.A. (1999). Validity Issues with the Family Environment Scale: Psychometric Resolution and Research Application with Alcoholic Families. Psychological Assessment, 11(3),315‑325.</ref> |Culture & Environment |Family |Parent |ce_p_fes |abcd_sscep01; fes02 | | | | |- |Parental Monitoring Survey<ref name=":30" /><ref name=":31" /> |Culture & Environment |Parenting |Parent |ce_p_pm |N/A | | | | |- |School Attendance of Youth & Grades<ref name=":34" /> |Culture & Environment |School |Parent |ce_p_sag |abcd_saag01 | | | | |- |Prosocial Behavior Survey - Parent<ref name=":27" /> |Culture & Environment |Temperament/Personality |Parent |ce_p_psb |psb01; abcd_sscep01 | | | | |- |HOME Short Form Cognitive Stimulation<ref>Bailey, C.T. & Boykin, A.W. (2001). The role of task variability and home contextual factors in the academic performance and task motivation of African American elementary school children. ''The Journal of Negro Education, 70''(1/2), 84-95. <nowiki>http://www.jstor.org/stable/2696285</nowiki></ref><ref>Boykin, A.W. & Cunningham, R.T. The effects of movement expressiveness in story content and learning context on the analogical reasoning performance of African American Children. ''Negro Education, 70''(1/2), 72-83. <nowiki>http://www.jstor.org/stable/2696284</nowiki></ref><ref>Zaslow, M. J., Weinfield, N. S., Gallagher, M., Hair, E. C., Ogawa, J. R., Egeland, B., ... & De Temple, J. M. (2006). Longitudinal prediction of child outcomes from differing measures of parenting in a low-income sample. ''Developmental Psychology'', ''42''(1), 27-37. <nowiki>https://doi.org/10.1037/0012-1649.42.1.27</nowiki></ref> |Culture & Environment | |Parent | | | | | | |- |Driving |Culture & Environment | |Parent | | | | | | |- | | | | | | | | | | |- |Fitbit - Physical Activity (Daily) |Novel Technologies |Actigraphy |Youth |nt_y_fitb_act_d |abcd_fbdpas01 | | | | |- |Fitbit - Physical Activity (Weekly) |Novel Technologies |Actigraphy |Youth |nt_y_fitb_act_w |abcd_fbdpas01; abcd_fbwpas01 | | | | |- |Fitbit - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Actigraphy |Youth |nt_y_fitb_qtn_plt |abcd_yfb01; abcd_yff01 | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Youth |nt_y_fitb_qtn |abcd_fbpay01 | | | | |- |Fitbit - Sleep (Daily) |Novel Technologies |Actigraphy |Youth |nt_y_fitb_slp_d |abcd_fbdpas01; abcd_fbdss01 | | | | |- |Fitbit - Sleep (Weekly) |Novel Technologies |Actigraphy |Youth |nt_y_fitb_slp_w |abcd_fbwss01; abcd_fbdss01; abcd_fbdpas01 | | | | |- |EARS - Device Usage Statistics (Pilot) |Novel Technologies |Screen Use |Youth |nt_y_ears_plt |abcd_mte01 | | | | |- |EARS - Device Usage Statistics |Novel Technologies |Screen Use |Youth |nt_y_ears |N/A | | | | |- |EARS - Post-Assessment Survey |Novel Technologies |Screen Use |Youth |nt_y_ears_qtn |N/A | | | | |- |EARS/Vibrent - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Screen Use |Youth |nt_y_ears_vibr_qtn_plt |abcd_mtpry01; abcd_mtpay01 | | | | |- |Screen Time Questionnaire |Novel Technologies |Screen Use |Youth |nt_y_st |abcd_ssmty01; abcd_stq01 | | | | |- |Vibrent - Device Usage Statistics (Pilot) |Novel Technologies |Screen Use |Youth |nt_y_vibr_plt |abcd_mtv01 | | | | |- | | | | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Actigraphy |Parent |nt_p_fitb_qtn_plt |abcd_pfb01, abcd_pff01 | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Parent |nt_p_fitb_qtn |abcd_fbprp01; abcd_fbpap01 | | | | |- |EARS - Post-Assessment Survey |Novel Technologies |Screen Use |Parent |nt_p_ears_qtn |N/A | | | | |- |EARS - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Screen Use |Parent |nt_p_ears_qtn_plt |abcd_mtpa01; abcd_mtpap01 | | | | |- |Parent Screen Time Questionnaire |Novel Technologies |Screen Use |Parent |nt_p_psq |screentime_psq_p01 | | | | |- |Screen Time Questionnaire (Parent) |Novel Technologies |Screen Use |Parent |nt_p_stq |stq01 | | | | |- | | | | | | | | | | |- |School Records |Other |School Records |Other | | | | | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Core - Imaging Data Coding Information !Measure !Category !Subcategory !Source !Table Name !Table Name (NDA 4.0) !SAS Code !SPSS Code !R Code |- |MRI Info |Brain Imaging |Administrative |Youth |mri_y_adm_info |abcd_mri01 | | | |- |Pre/Post-Scan Questionnaires |Brain Imaging |Administrative |Youth |mri_y_adm_qtn |abcd_ypsq201; abcd_ypre201; abcd_ypre101 | | | |- |Scanning Checklist and Notes |Brain Imaging |Administrative |Youth |mri_y_adm_nts |abcd_ra01 | | | |- |Fractional Anisotropy (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_at |abcd_dmdtifp101 | | | |- |Fractional Anisotropy (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_aseg |abcd_dmdtifp101 | | | |- |Fractional Anisotropy - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_gm_dsk |abcd_dmdtifp101 | | | |- |Fractional Anisotropy - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_gm_dst |abcd_ddtifp101 | | | |- |Fractional Anisotropy - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_gwc_dsk |abcd_dmdtifp101; abcd_dmdtifp202 | | | |- |Fractional Anisotropy - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_gwc_dst |abcd_ddtifp201 | | | |- |Fractional Anisotropy - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_wm_dsk |abcd_dmdtifp101 | | | |- |Fractional Anisotropy - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_fa_fs_wm_dst |abcd_ddtifp101 | | | |- |Longitudinal Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_at |abcd_dmdtifp101 | | | |- |Longitudinal Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_aseg |abcd_dmdtifp101 | | | |- |Longitudinal Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_gm_dsk |abcd_dmdtifp101 | | | |- |Longitudinal Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_gm_dst |abcd_ddtifp101; abcd_ddtifp201 | | | |- |Longitudinal Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_gwc_dsk |abcd_dmdtifp202 | | | |- |Longitudinal Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_gwc_dst |abcd_ddtifp201 | | | |- |Longitudinal Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_wm_dsk |abcd_dmdtifp101 | | | |- |Longitudinal Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_ld_fs_wm_dst |abcd_ddtifp101 | | | |- |Mean Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_at |abcd_dmdtifp101 | | | |- |Mean Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_aseg |abcd_dmdtifp101 | | | |- |Mean Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_gm_dsk |abcd_dmdtifp101 | | | |- |Mean Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_gm_dst |abcd_ddtifp101 | | | |- |Mean Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_gwc_dsk |abcd_dmdtifp202 | | | |- |Mean Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_gwc_dst |abcd_ddtifp201 | | | |- |Mean Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_wm_dsk |abcd_dmdtifp101 | | | |- |Mean Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_md_fs_wm_dst |abcd_ddtifp101 | | | |- |Transverse Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_at |abcd_dmdtifp101 | | | |- |Transverse Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_aseg |abcd_dmdtifp101 | | | |- |Transverse Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_gm_dsk |abcd_dmdtifp101 | | | |- |Transverse Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_gm_dst |abcd_ddtifp201 | | | |- |Transverse Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_gwc_dsk |abcd_dmdtifp202 | | | |- |Transverse Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_gwc_dst |abcd_ddtifp201 | | | |- |Transverse Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_wm_dsk |abcd_dmdtifp101 | | | |- |Transverse Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_td_fs_wm_dst |abcd_ddtifp101 | | | |- |Volume (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Full Shell) |Youth |mri_y_dti_vol_fs_at |abcd_dmdtifp101 | | | |- |Fractional Anisotropy (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_at |abcd_dti_p101 | | | |- |Fractional Anisotropy (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_aseg |abcd_dti_p101 | | | |- |Fractional Anisotropy - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_gm_dsk |abcd_dti_p101 | | | |- |Fractional Anisotropy - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_gm_dst |abcd_ddtidp101 | | | |- |Fractional Anisotropy - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_gwc_dsk |abcd_dti_p201 | | | |- |Fractional Anisotropy - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_gwc_dst |abcd_ddtidp201 | | | |- |Fractional Anisotropy - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_wm_dsk |abcd_dti_p101 | | | |- |Fractional Anisotropy - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_fa_is_wm_dst |abcd_ddtidp101 | | | |- |Longitudinal Diffusivity (Atlas Track) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_at |abcd_dti_p101 | | | |- |Longitudinal Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_aseg |abcd_dti_p101 | | | |- |Longitudinal Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_gm_dsk |abcd_dti_p201 | | | |- |Longitudinal Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_gm_dst |abcd_ddtidp101; abcd_ddtidp201 | | | |- |Longitudinal Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_gwc_dsk |abcd_dti_p201 | | | |- |Longitudinal Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_gwc_dst |abcd_ddtidp201 | | | |- |Longitudinal Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_wm_dsk |abcd_dti_p101 | | | |- |Longitudinal Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_ld_is_wm_dst |abcd_ddtidp101 | | | |- |Mean Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_at |abcd_dti_p101 | | | |- |Mean Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_aseg |abcd_dti_p101 | | | |- |Mean Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_gm_dsk |abcd_dti_p101 | | | |- |Mean Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_gm_dst |abcd_ddtidp101 | | | |- |Mean Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_gwc_dsk |abcd_dti_p201 | | | |- |Mean Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_gwc_dst |abcd_ddtidp201 | | | |- |Mean Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_wm_dsk |abcd_dti_p101 | | | |- |Mean Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_md_is_wm_dst |abcd_ddtidp101 | | | |- |Transverse Diffusivity (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_at |abcd_dti_p101 | | | |- |Transverse Diffusivity (Subcortical) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_aseg |abcd_dti_p101 | | | |- |Transverse Diffusivity - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_gm_dsk |abcd_dti_p201 | | | |- |Transverse Diffusivity - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_gm_dst |abcd_ddtidp201 | | | |- |Transverse Diffusivity - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_gwc_dsk |abcd_dti_p201 | | | |- |Transverse Diffusivity - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_gwc_dst |abcd_ddtidp201 | | | |- |Transverse Diffusivity - White Matter (Desikan) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_wm_dsk |abcd_dti_p101 | | | |- |Transverse Diffusivity - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_td_is_wm_dst |abcd_ddtidp101 | | | |- |Volume (AtlasTrack) |Brain Imaging |Diffusion MRI (DTI Inner Shell) |Youth |mri_y_dti_vol_is_at |abcd_dti_p101 | | | |- |Free Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_at |abcd_drsip701 | | | |- |Free Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_aseg |abcd_drsip701 | | | |- |Free Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_gm_dsk |abcd_drsip701 | | | |- |Free Normalized Isotropic - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_gm_dst |abcd_drsip701 | | | |- |Free Normalized Isotropic - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_gwc_dsk |abcd_drsip701 | | | |- |Free Normalized Isotropic - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_gwc_dst |abcd_drsip701 | | | |- |Free Normalized Isotropic - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_wm_dsk |abcd_drsip701 | | | |- |Free Normalized Isotropic - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_fni_wm_dst |abcd_drsip701 | | | |- |Hindered Normalized Directional (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_at |abcd_drsip501 | | | |- |Hindered Normalized Directional (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_aseg |abcd_drsip501 | | | |- |Hindered Normalized Directional - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_gm_dsk |abcd_drsip501 | | | |- |Hindered Normalized Directional - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_gm_dst |abcd_drsip501 | | | |- |Hindered Normalized Directional - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_gwc_dsk |abcd_drsip501 | | | |- |Hindered Normalized Directional - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_gwc_dst |abcd_drsip501 | | | |- |Hindered Normalized Directional - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_wm_dsk |abcd_drsip501 | | | |- |Hindered Normalized Directional - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnd_wm_dst |abcd_drsip501 | | | |- |Hindered Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_at |abcd_drsip401 | | | |- |Hindered Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_aseg |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_gm_dsk |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_gm_dst |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_gwc_dsk |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_gwc_dst |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_wm_dsk |abcd_drsip401 | | | |- |Hindered Normalized Isotropic - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hni_wm_dst |abcd_drsip401 | | | |- |Hindered Normalized Total (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_at |abcd_drsip601 | | | |- |Hindered Normalized Total (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_aseg |abcd_drsip601 | | | |- |Hindered Normalized Total - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_gm_dsk |abcd_drsip601 | | | |- |Hindered Normalized Total - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_gm_dst |abcd_drsip601 | | | |- |Hindered Normalized Total - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_gwc_dsk |abcd_drsip601 | | | |- |Hindered Normalized Total - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_gwc_dst |abcd_drsip601 | | | |- |Hindered Normalized Total - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_wm_dsk |abcd_drsip601 | | | |- |Hindered Normalized Total - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_hnt_wm_dsk |abcd_drsip601 | | | |- |Restricted Normalized Directional (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_at |abcd_drsip201 | | | |- |Restricted Normalized Directional (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_aseg |abcd_drsip201 | | | |- |Restricted Normalized Directional - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_gm_dsk |abcd_drsip201 | | | |- |Restricted Normalized Directional - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_gm_dst |abcd_drsip201 | | | |- |Restricted Normalized Directional - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_gwc_dsk |abcd_drsip201 | | | |- |Restricted Normalized Directional - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_gwc_dst |abcd_drsip201 | | | |- |Restricted Normalized Directional - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_wm_dsk |abcd_drsip201 | | | |- |Restricted Normalized Directional - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnd_wm_dst |abcd_drsip201 | | | |- |Restricted Normalized Isotropic (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_at |abcd_drsip101 | | | |- |Restricted Normalized Isotropic (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_aseg |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_gm_dsk |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_gm_dst |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_gwc_dsk |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_gwc_dst |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_wm_dsk |abcd_drsip101 | | | |- |Restricted Normalized Isotropic - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rni_wm_dst |abcd_drsip101 | | | |- |Restricted Normalized Total (AtlasTrack) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_at |abcd_drsip301 | | | |- |Restricted Normalized Total (Subcortical) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_aseg |abcd_drsip301 | | | |- |Restricted Normalized Total - Gray Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_gm_dsk |abcd_drsip301 | | | |- |Restricted Normalized Total - Gray Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_gm_dst |abcd_drsip301 | | | |- |Restricted Normalized Total - Gray/White Contrast (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_gwc_dsk |abcd_drsip301 | | | |- |Restricted Normalized Total - Gray/White Contrast (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_gwc_dst |abcd_drsip301 | | | |- |Restricted Normalized Total - White Matter (Desikan) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_wm_dsk |abcd_drsip301 | | | |- |Restricted Normalized Total - White Matter (Destrieux) |Brain Imaging |Diffusion MRI (RSI) |Youth |mri_y_rsi_rnt_wm_dst |abcd_drsip301 | | | |- |Resting State fMRI - Correlations (Gordon Network to Subcortical) |Brain Imaging |Resting State fMRI |Youth |mri_y_rsfmr_cor_gp_aseg |mrirscor02 | | | |- |Resting State fMRI - Correlations (Gordon Network) |Brain Imaging |Resting State fMRI |Youth |mri_rsfmr_cor_gp_gp |abcd_betnet02 | | | |- |Resting State fMRI - Temporal Variance (Desikan) |Brain Imaging |Resting State fMRI |Youth |mri_y_rsfmr_var_dsk |abcd_mrirstv02 | | | |- |Resting State fMRI - Temporal Variance (Destrieux) |Brain Imaging |Resting State fMRI |Youth |mri_y_rsfmr_var_dst |abcd_mrirsfd01 | | | |- |Resting State fMRI - Temporal Variance (Gordon) |Brain Imaging |Resting State fMRI |Youth |mri_y_rsfmr_var_gp |abcd_mrirstv02 | | | |- |Resting State fMRI - Temporal Variance (Subcortical) |Brain Imaging |Resting State fMRI |Youth |mri_y_rsfmr_var_aseg |abcd_mrirstv02 | | | |- |Cortical Thickness (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_thk_dsk |abcd_smrip102 | | | |- |Cortical Thickness (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_thk_dst |abcd_mrisdp102 | | | |- |Cortical Thickness (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_thk_fzy |abcd_smrip102 | | | |- |Sulcal Depth (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_sulc_dsk |abcd_smrip102 | | | |- |Sulcal Depth (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_sulc_dst |abcd_mrisdp102 | | | |- |Sulcal Depth (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_sulc_fzy |abcd_smrip102 | | | |- |Surface Area (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_area_dsk |abcd_smrip102 | | | |- |Surface Area (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_area_dst |abcd_mrisdp102 | | | |- |Surface Area (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_area_fzy |abcd_smrip102 | | | |- |T1 Intensity (Subcortical) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_aseg |abcd_smrip202 | | | |- |T1 Intensity - Gray Matter (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_gray_dsk |abcd_smrip202 | | | |- |T1 Intensity - Gray Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_gray_dst |abcd_mrisdp202 | | | |- |T1 Intensity - Gray Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_gray_fzy |abcd_smrip202 | | | |- |T1 Intensity - Gray/White Contrast (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_contr_dsk |abcd_smrip202 | | | |- |T1 Intensity - Gray/White Contrast (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_contr_dst |abcd_mrisdp202 | | | |- |T1 Intensity - Gray/White Contrast (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_contr_fzy |abcd_smrip202 | | | |- |T1 Intensity - White Matter (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_white_dsk |abcd_smrip202 | | | |- |T1 Intensity - White Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_white_dst |abcd_mrisdp202 | | | |- |T1 Intensity - White Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t1_white_fzy |abcd_smrip202 | | | |- |T2 Intensity (Subcortical) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_aseg |abcd_smrip302 | | | |- |T2 Intensity - Gray Matter (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_gray_dsk |abcd_smrip302 | | | |- |T2 Intensity - Gray Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_gray_dst |abcd_mrisdp302 | | | |- |T2 Intensity - Gray Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_gray_fzy |abcd_smrip302 | | | |- |T2 Intensity - Gray/White Contrast (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_contr_dsk |abcd_smrip302 | | | |- |T2 Intensity - Gray/White Contrast (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_contr_dst |abcd_mrisdp302 | | | |- |T2 Intensity - Gray/White Contrast (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_contr_fzy |abcd_smrip302 | | | |- |T2 Intensity - White Matter (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_white_dsk |abcd_smrip302 | | | |- |T2 Intensity - White Matter (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_white_dst |abcd_mrisdp302 | | | |- |T2 Intensity - White Matter (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_t2_white_fzy |abcd_smrip302 | | | |- |Volume (Desikan) |Brain Imaging |Structural MRI |Youth |mri_y_smr_vol_dsk |abcd_smrip102 | | | |- |Volume (Destrieux) |Brain Imaging |Structural MRI |Youth |mri_y_smr_vol_dst |abcd_mrisdp102 | | | |- |Volume (Fuzzy Clustering) |Brain Imaging |Structural MRI |Youth |mri_y_smr_vol_fzy |abcd_smrip102 | | | |- |Volume (Subcortical) |Brain Imaging |Structural MRI |Youth |mri_y_smr_vol_aseg |abcd_smrip102 | | | |- |Anticipated large loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref name=":43">Knutson B, Westdorp A, Kaiser E, Hommer D (2000) FMRI visualization of brain activity during a monetary incentive delay task. NeuroImage 12: 20–27.</ref> |Youth |mri_y_tfmr_mid_allvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated large loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_allvn_dst |abcd_midabwdp202; abcd_midasemdp202; abcd_midr1bwdp202; abcd_tr2semdp202 | | | |- |Anticipated large loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_allvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated large reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alrvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated large reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alrvn_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; abcd_tr2semdp101 | | | |- |Anticipated large reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alrvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated large vs. small loss (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsl_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated large vs. small loss (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsl_dst |abcd_midabwdp202; abcd_midasemdp202; abcd_midr1bwdp202; abcd_tmidr1semdp202; abcd_tr2bwdp202; abcd_tr2semdp202 | | | |- |Anticipated large vs. small loss (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsl_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated large vs. small reward (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsr_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated large vs. small reward (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsr_dst |abcd_midabwdp01; abcd_midabwdp202; abcd_midasemdp101; abcd_midasemdp202; abcd_midr1bwdp101; abcd_midr1bwdp202; abcd_tmidr1semdp101; abcd_tmidr1semdp202; abcd_tr2bwdp01; abcd_tr2bwdp202; abcd_tr2semdp101; abcd_tr2semdp202 | | | |- |Anticipated large vs. small reward (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvsr_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102; | | | |- |Anticipated loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202; | | | |- |Anticipated loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvn_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; abcd_tr2semdp101 | | | |- |Anticipated loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_alvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_arvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_arvn_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; abcd_tr2semdp101 | | | |- |Anticipated reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_arvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated small loss vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_aslvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated small loss vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_aslvn_dst |abcd_midabwdp202; abcd_midasemdp202; abcd_midr1bwdp202; abcd_tmidr1semdp202; abcd_tr2bwdp202; abcd_tr2semdp202 | | | |- |Anticipated small loss vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_aslvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Anticipated small reward vs. neutral (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_asrvn_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Anticipated small reward vs. neutral (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_asrvn_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; bcd_tr2semdp101 | | | |- |Anticipated small reward vs. neutral (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_asrvn_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Behavioral Performance |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_beh |abcd_mid02 | | | |- |Loss positive vs. negative feedback (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_lpvnf_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Loss positive vs. negative feedback (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_lpvnf_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; abcd_tr2semdp101 | | | |- |Loss positive vs. negative feedback (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_lpvnf_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Post-Scan Questionnaire |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_qtn |abcd_monet01; abcd_prepost01 | | | |- |Reward positive vs. negative feedback (Desikan) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_rpvnf_dsk |abcd_midasemp202; abcd_midr1bwp202; abcd_midr2semp202; abcd_midsemp202; midaparcp203; midr2bwp202 | | | |- |Reward positive vs. negative feedback (Destrieux) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_rpvnf_dst |abcd_midabwdp01; abcd_midasemdp101; abcd_midr1bwdp101; abcd_tmidr1semdp101; abcd_tr2bwdp01; abcd_tr2semdp101 | | | |- |Reward positive vs. negative feedback (Subcortical) |Brain Imaging |Task fMRI - Monetary Incentive Delay Task<ref name=":5" /><ref name=":43" /> |Youth |mri_y_tfmr_mid_rpvnf_aseg |abcd_midasemp102; abcd_midr1bwp102; abcd_midr2semp102; abcd_midsemp102; midaparc03; midr2bwp102 | | | |- |Recognition memory behavioral performance |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44">Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |mri_y_tfmr_nback_rec_beh |mribrec02 | | | |- |0-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_0b_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |0-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_0b_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |0-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_0b_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |2-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_2b_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |2-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_2b_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |2-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":5" /><ref name=":44" /> |Youth |mri_y_tfmr_nback_2b_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |2-back vs. 0-back (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":52">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |mri_y_tfmr_nback_2bv0b_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |2-back vs. 0-back (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":53">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |mri_y_tfmr_nback_2bv0b_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |2-back vs. 0-back (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":54">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref>Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |mri_y_tfmr_nback_2bv0b_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Behavioral performance |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55">ABCD Study. (n.d.). ''fMRI Tasks and Tools''. Retrieved August 5, 2024, from <nowiki>https://abcdstudy.org/scientists/abcd-fmri-tasks-and-tools/</nowiki></ref><ref name=":45">Cohen, A.O., Conley, M.I., Dellarco, D.V., Casey, B.J. (November, 2016). The impact of emotional cues on short-term and long-term memory during adolescence. Society for Neuroscience, San Diego, CA.</ref> |Youth |mri_y_tfmr_nback_beh |abcd_mrinback02 | | | |- |Emotion (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emo_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Emotion (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emo_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |Emotion (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emo_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Emotion vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emovntf_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Emotion vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emovntf_dst |abcd_tfabwdp101; abcd_tfabwdp201; abcd_tfnbr1semdp101; abcd_tfnbr1semdp201; abcd_tfnbr2bwdp101; abcd_tfnbr2bwdp201; abcd_tfnbr2dp101; abcd_tfnbr2dp201; abcd_tfncr1bwdp101; abcd_tfncr1bwdp201; abcd_tnbasemdp101; abcd_tnbasemdp201 | | | |- |Emotion vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_emovntf_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Face vs. place (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_fvplc_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Face vs. place (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_fvplc_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |Face vs. place (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_fvplc_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Negative face vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_ngfvntf_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Negative face vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_ngfvntf_dst |abcd_tfabwdp201; abcd_tfnbr1semdp201; abcd_tfnbr2bwdp201; abcd_tfnbr2dp201; abcd_tfncr1bwdp201; abcd_tnbasemdp201 | | | |- |Negative face vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_ngfvntf_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Place (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_plc_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Place (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_plc_dst |abcd_tfabwdp101; abcd_tfnbr1semdp101; abcd_tfnbr2bwdp101; abcd_tfnbr2dp101; abcd_tfncr1bwdp101; abcd_tnbasemdp101 | | | |- |Place (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_plc_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Positive face vs. neutral face (Desikan) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_psfvntf_dsk |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Positive face vs. neutral face (Destrieux) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_psfvntf_dst |abcd_tfabwdp201; abcd_tfnbr1semdp201; abcd_tfnbr2bwdp201; abcd_tfnbr2dp201; abcd_tfncr1bwdp201; abcd_tnbasemdp201 | | | |- |Positive face vs. neutral face (Subcortical) |Brain Imaging |Task fMRI - Emotional N-Back Task<ref name=":55" /><ref name=":45" /> |Youth |mri_y_tfmr_nback_psfvntf_aseg |nback_bwroi02; nbackallsem01; nbackr101; nbackr1sem01; nbackr201; nbackr2sem01 | | | |- |Behavioral performance |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46">Logan GD (1994) On the ability to inhibit thought and action: A users’ guide to the stop signal paradigm. In D. Dagenbach & T. H. Carr (Eds), Inhibitory processes in attention, memory, and language: 189-239. San Diego: Academic Press</ref> |Youth |mri_y_tfmr_sst_beh |abcd_sst02 | | | |- |Any stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_asvcg_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Any stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_asvcg_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Any stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_asvcg_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Correct go vs. fixation (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_cgvfx_dsk |mrisst02; mrisstsem01; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01 | | | |- |Correct go vs. fixation (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_cgvfx_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Correct go vs. fixation (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_cgvfx_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Correct stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvcg_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Correct stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvcg_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Correct stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvcg_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Correct stop vs. incorrect stop (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvis_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Correct stop vs. incorrect stop (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvis_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Correct stop vs. incorrect stop (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_csvis_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect go vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvcg_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect go vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvcg_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Incorrect go vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvcg_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect go vs. incorrect stop (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvis_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect go vs. incorrect stop (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvis_dst |abcd_tfsstabwdp101; abcd_tfsstabwdp201; abcd_tfsstasemdp101; abcd_tfsstasemdp201; abcd_tfsstr1bwdp101; abcd_tfsstr1bwdp201; abcd_tfsstr1semdp101; abcd_tfsstr1semdp201; abcd_tfsstr2bwdp101; abcd_tfsstr2bwdp201; abcd_tfsstr2semdp101; abcd_tfsstr2semdp201 | | | |- |Incorrect go vs. incorrect stop (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_igvis_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect stop vs. correct go (Desikan) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_isvcg_dsk |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Incorrect stop vs. correct go (Destrieux) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_isvcg_dst |abcd_tfsstabwdp101; abcd_tfsstasemdp101; abcd_tfsstr1bwdp101; abcd_tfsstr1semdp101; abcd_tfsstr2bwdp101; abcd_tfsstr2semdp101 | | | |- |Incorrect stop vs. correct go (Subcortical) |Brain Imaging |Task fMRI - Stop Signal Task<ref name=":5" /><ref name=":46" /> |Youth |mri_y_tfmr_sst_isvcg_aseg |mrisst02; mrisstr1bw01; mrisstr1sem01; mrisstr2bw01; mrisstr2bwsem01; mrisstsem01 | | | |- |Automatic - Post-processing |Brain Imaging |Quality Control |Youth |mri_y_qc_auto_post |abcd_auto_postqc01 | | | |- |MRI Clinical Report/FIndings |Brain Imaging |Quality Control |Youth |mri_y_qc_clfind |abcd_mrfindings02 | | | |- |Manual - Freesurfer |Brain Imaging |Quality Control |Youth |mri_y_qc_man_fsurf |abcd_fsurfqc01 | | | |- |Manual - Post-processing - Diffusion MRI |Brain Imaging |Quality Control |Youth |mri_y_qc_man_post_dmr |abcd_dmriqc01 | | | |- |Manual - Post-processing - Functional MRI |Brain Imaging |Quality Control |Youth |mri_y_qc_man_post_fmr |abcd_fmriqc01 | | | |- |Manual - Post-processing - Structural MRI - T2w |Brain Imaging |Quality Control |Youth |mri_y_qc_man_post_t2w |abcd_t2wqc01 | | | |- |Motion |Brain Imaging |Quality Control |Youth |mri_y_qc_motion |abcd_betnet02; abcd_dmdtifp202; abcd_midasemp102; mrisst02; mrisstr1bw01; mrisstr2bw01; nback_bwroi02; nbackr101; nbackr201 | | | |- |Raw - Diffusion MRI |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_dmr |mriqcrp103; mriqcrp302 | | | |- |Raw - Event |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_event |mriqcrp302 | | | |- |Raw - Resting State fMRI |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_rsfmr |mriqcrp103; mriqcrp302 | | | |- |Raw - Structural MRI - T1 |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_smr_t1 |mriqcrp103; mriqcrp302 | | | |- |Raw - Structural MRI - T2 |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_smr_t2 |mriqcrp103; mriqcrp302 | | | |- |Raw - Task fMRI - All |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_tfmr_all |mriqcrp103; mriqcrp302 | | | |- |Raw - Task fMRI - MID |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_tfmr_mid |mriqcrp103; mriqcrp203; mriqcrp302 | | | |- |Raw - Task fMRI - N-Back |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_tfmr_nback |mriqcrp203; mriqcrp302 | | | |- |Raw - Task fMRI - SST |Brain Imaging |Quality Control |Youth |mri_y_qc_raw_tfmr_sst |mriqcrp203; mriqcrp302 | | | |- |Recommended Image Inclusion |Brain Imaging |Quality Control |Youth |mri_y_qc_incl |abcd_imgincl01 | | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Core - Linked Data Coding Information !Measure !Category !Subcategory !Source !Table Name !Table Name (NDA 4.0) !Subscale Information !SAS Code !SPSS Code !R Code |- |Geocoding from Residential History |Linked Data |Administrative |Other |led_l_admin |abcd_rhds01 | | | | |- |Satellite-based NO2 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |led_l_no2 |abcd_rhds01 | | | | |- |Satellite-based O3 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |led_l_o3 |abcd_rhds01 | | | | |- |Satellite-based PM2.5 Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |led_l_pm25 |abcd_rhds01 | | | | |- |Satellite-based Particulate Measures |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |led_l_particulat |N/A | | | | |- |Satellite-based Pollution Measures for Prenatal Addresses |Linked Data |Air Pollution |Air Quality Data for Health-Related Applications |led_l_prenatal |N/A | | | | |- |Neighborhood SES and Demographics |Linked Data |Amenities & Services |NaDA |led_l_nbhsoc |N/A | | | | |- |Parks |Linked Data |Amenities & Services |NaDA |led_l_parks |N/A | | | | |- |Performing Arts and Sports Recreation Orgs |Linked Data |Amenities & Services |NaDA |led_l_artsports |N/A | | | | |- |Religious/Civic Organizations |Linked Data |Amenities & Services |NaDA |led_l_relciv |N/A | | | | |- |Social Service |Linked Data |Amenities & Services |NaDA |led_l_socsrv |N/A | | | | |- |Building Density |Linked Data |Built Environment |EPA |led_l_densbld |abcd_rhds01 | | | | |- |Crime |Linked Data |Built Environment |ICPSR |led_l_crime |abcd_rhds01 | | | | |- |Lead Risk |Linked Data |Built Environment |Vox |led_l_leadrisk |abcd_rhds01 | | | | |- |Population Density |Linked Data |Built Environment |EPA |led_l_denspop |abcd_rhds01 | | | | |- |Road Proximity |Linked Data |Built Environment |Kalibrate |led_l_roadprox |abcd_rhds01 | | | | |- |Traffic Density |Linked Data |Built Environment |Kalibrate |led_l_traffic |abcd_rhds01 | | | | |- |Urban/Rural Area |Linked Data |Built Environment |Census |led_l_urban |abcd_rhds01 | | | | |- |Vehicle Density |Linked Data |Built Environment |ACS |led_l_densveh |N/A | | | | |- |Walkability |Linked Data |Built Environment |EPA |led_l_walk |abcd_rhds01 | | | | |- |Behavioral Health Measures |Linked Data |Community Health Burden |PLACES |led_l_places |N/A | | | | |- |Elevation of Address |Linked Data |Meteorology & Exposures |Google API |led_l_elevation |abcd_rhds01 | | | | |- |Estimates of Environmental Noise |Linked Data |Meteorology & Exposures |Harvard |led_l_noise |N/A | | | | |- |Selected EJScreen Measures |Linked Data |Meteorology & Exposures |EJScreen |led_l_ejscreen |N/A | | | | |- |Temperature Estimates |Linked Data |Meteorology & Exposures |PRISM |led_l_temp |abcd_rhds01 | | | | |- |VPD Estimates |Linked Data |Meteorology & Exposures |PRISM |led_l_vpd |abcd_rhds01 | | | | |- |Land-use Measures |Linked Data |Natural Space & Satellite |NLT |led_l_urbsat |N/A | | | | |- |Measure of Land Cover and Tree Canopy |Linked Data |Natural Space & Satellite |NLCD |led_l_nlcd |N/A | | | | |- |Alcohol Outlet Density |Linked Data |Neighborhood Social Factors |Census 2016 |led_l_densalc |N/A | | | | |- |Anomie/Disenfranchisement/Social Capital |Linked Data |Neighborhood Social Factors |Census Return |led_l_censusret |N/A | | | | |- |Number of Jobs and Job Density |Linked Data |Neighborhood Social Factors |LODES |led_l_lodes |N/A | | | | |- |Opportunity Zones and Investment Scores (OZ) |Linked Data |Neighborhood Social Factors |Other |led_l_oz |N/A | | | | |- |Rent and Mortgage Statistics |Linked Data |Neighborhood Social Factors |ACS |led_l_rentmort |N/A | | | | |- |Social Mobility |Linked Data |Neighborhood Social Factors |Opportunity Atlas |led_l_socmob |abcd_rhds01 | | | | |- |Area Deprivation Index (ADI) |Linked Data |Neighborhood Composite Measures |Other |led_l_adi |abcd_rhds01 | | | | |- |Child Opportunity Index 2.0 (COI) |Linked Data |Neighborhood Composite Measures |Other |led_l_coi |abcd_rhds01 | | | | |- |Minority Health Social Vulnerability Index (MHSVI) |Linked Data |Neighborhood Composite Measures |Other |led_l_mhsvi | | | | | |- |Social Vulnerability Index (SVI) |Linked Data |Neighborhood Composite Measures |Other |led_l_svi |abcd_rhds01 | | | | |- |Affordable Care Act Medicaid Expansion Data |Linked Data |Policy Vars |KFF |led_l_aca | | | | | |- |CDC Opioid Prescription Dispensing Data per 100k Residents |Linked Data |Policy Vars |CDC |led_l_rxopioid | | | | | |- |Cannabis Legalizations Categories by State |Linked Data |Policy Vars |NCSL and MPP |led_l_lawsmj |abcd_rhds01 | | | | |- |Gender Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |led_l_biasgender |abcd_rhds01 | | | | |- |Immigration Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |led_l_biasimm |abcd_rhds01 | | | | |- |OPTIC-Vetted Co-prescribing Naloxone Policy Data |Linked Data |Policy Vars |OPTIC |led_l_rxnalox |N/A | | | | |- |OPTIC-Vetted Good Samaritan Policy Data |Linked Data |Policy Vars |OPTIC |led_l_goodsam |N/A | | | | |- |OPTIC-Vetted Medical Marijuana Policy Data |Linked Data |Policy Vars |OPTIC |led_l_medmj |N/A | | | | |- |OPTIC-Vetted Naloxone Policy Data |Linked Data |Policy Vars |OPTIC |led_l_polnalox |N/A | | | | |- |OPTIC-Vetted Prescription Drug Monitoring Program Policy Data (PDMP) |Linked Data |Policy Vars |OPTIC |led_l_rxmonit |N/A | | | | |- |Race Bias Measures |Linked Data |Policy Vars |Hatzenbuehler |led_l_biasrace |abcd_rhds01 | | | | |- |Dissimilarity Index |Linked Data |Residential Segregation |ACS |led_l_dissim |N/A | | | | |- |Exposure/Interaction Index |Linked Data |Residential Segregation |ACS |led_l_expint |N/A | | | | |- |GI Statistics |Linked Data |Residential Segregation |ICPSR |led_l_gi |N/A | | | | |- |Index of Concentration at the Extremes |Linked Data |Residential Segregation |ACS |led_l_ice |N/A | | | | |- |Multi-Group Entropy Index |Linked Data |Residential Segregation |ACS |led_l_entropy |N.A | | | | |- |County |Linked Data |School (Demographics) |SEDA |led_l_seda_demo_c |led_school_part_301 | | | | |- |District |Linked Data |School (Demographics) |SEDA |led_l_seda_demo_d |led_school_part_201 | | | | |- |Metro Area |Linked Data |School (Demographics) |SEDA |led_l_seda_demo_m |led_school_part_501 | | | | |- |School |Linked Data |School (Demographics) |SEDA |led_l_seda_demo_s |led_school_part_101 | | | | |- |Commuting Zone |Linked Data |School (Math & Reading) |SEDA |led_l_seda_pool_z |led_school_part_401 | | | | |- |County |Linked Data |School (Math & Reading) |SEDA |led_l_seda_pool_c |led_school_part_301 | | | | |- |District |Linked Data |School (Math & Reading) |SEDA |led_l_seda_pool_d |led_school_part_201 | | | | |- |Metro Area |Linked Data |School (Math & Reading) |SEDA |led_l_seda_pool_m |led_school_part_501 | | | | |- |School |Linked Data |School (Math & Reading) |SEDA |led_l_seda_pool_s |led_school_part_101 | | | | |- |Commuting Zone |Linked Data |School (Math Poolsub) |SEDA |led_l_seda_math_z |led_school_part_401 | | | | |- |County |Linked Data |School (Math Poolsub) |SEDA |led_l_seda_math_c |led_school_part_301 | | | | |- |District |Linked Data |School (Math Poolsub) |SEDA |led_l_seda_math_d |led_school_part_201 | | | | |- |Metro Area |Linked Data |School (Math Poolsub) |SEDA |led_l_seda_math_m |led_school_part_501 | | | | |- |Commuting Zone |Linked Data |School (Reading Poolsub) |SEDA |led_l_seda_read_z |led_school_part_401 | | | | |- |County |Linked Data |School (Reading Poolsub) |SEDA |led_l_seda_read_c |led_school_part_301 | | | | |- |District |Linked Data |School (Reading Poolsub) |SEDA |led_l_seda_read_d |led_school_part_201 | | | | |- |Metro Area |Linked Data |School (Reading Poolsub) |SEDA |led_l_seda_read_m |led_school_part_501 | | | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Substudy - Coding Information !Measure !Category !Subcategory !Source !Table Name !Table Name (NDA 4.0) !Subscale Information !SAS Code !SPSS Code !R Code |- |COVID-19 Fitbit Physical Activity (Daily) |COVID-19 |Actigraphy |Youth |cvd_y_fitb_act_d |N/A | | | | |- |COVID-19 Fitbit Physical Activity (Weekly) |COVID-19 |Actigraphy |Youth |cvd_y_fitb_act_w |N/A | | | | |- |COVID-19 Fitbit Post-Assessment Survey |COVID-19 |Actigraphy |Youth |cvd_y_fitb_qtn |covid19_fitbit_survey01 | | | | |- |COVID-19 Fitbit Sleep (Daily) |COVID-19 |Actigraphy |Youth |cvd_y_fitb_slp_d |N/A | | | | |- |COVID-19 Fitbit Sleep (Weekly) |COVID-19 |Actigraphy |Youth |cvd_y_fitb_slp_w |N/A | | | | |- |COVID-19 Questionnaire |COVID-19 |COVID |Youth |cvd_y_qtn |yabcdcovid19questionnaire01 | | | | |- |Endocannabinoid Substudy |Endocannabinoid |SU Consequence |Youth |ecb_y_ecb |N/A | | | | |- |Hurricane Irma Experiences |Hurricane Irma |Questionnaire |Youth |irma_y_qtn |abcd_isc01 | | | | |- |Reported Delinquency<ref name=":8">Elliott DS, Ageton SS, Huizinga D, Knowles BA, Canter RJ. ''The prevalence and incidence of delinquent behavior: 1976–1980 (National Youth Survey Report No. 26)'' Behavioral Research Institute; Boulder, CO: 1983.</ref> |Social Development |Delinquency |Youth |sd_y_rd |abcd_socdev_child_rde01; soc_dev_fu_rep_delinq01 | | | | |- |Difficulties in Emotion Regulation |Social Development |Emotion |Youth |sd_y_ders |abcd_socdev_child_emr01; soc_dev_fu_diff_emo_reg01 | | | | |- |Firearms (YRBSS) |Social Development |Firearm Storage |Youth |sd_y_fa |abcd_socdev_child_fa01; soc_dev_fu_firearms01 | | | | |- |Alabama Parenting Questionnaire |Social Development |Parenting |Youth |sd_y_apq |abcd_socdev_child_alabam01; soc_dev_fu_alabama01 | | | | |- |Peer Behavior |Social Development |Peers |Youth |sd_y_pb |abcd_socdev_child_pb01; soc_dev_fu_peer_behav01 | | | | |- |Personality Disposition |Social Development |Temperament/Personality |Youth |sd_y_pd |abcd_socdev_child_pdis01; soc_dev_fu_personality01; | | | | |- |Victimization |Social Development |Victimization |Youth |sd_y_vict |abcd_socdev_child_vic01; soc_dev_fu_victimize01 | | | | |- | | | | | | | | | | |- |COVID-19 Questionnaire |COVID-19 |COVID |Parent |cvd_p_qtn |pabcdcovid19questionnaire01 | | | | |- |Hurricane Irma Experiences |Hurricane Irma |Questionnaire |Parent |irma_p_qtn |abcd_ip01 | | | | |- |Visit Type |Social Development |Administrative |Parent |sd_p_vt |abcd_socdev_vt01; soc_dev_fu_visit_type01; | | | | |- |Perception of Neighborhood Scale |Social Development |Community |Parent |sd_p_nbh |soc_dev_fu_neighbor_p01; abcd_socdev_p_nbh01 | | | | |- |Reported Delinquency |Social Development |Delinquency |Parent |sd_p_rd |soc_dev_fu_rep_delinq_p01; abcd_socdev_p_rde01; | | | | |- |Difficulties in Emotion Regulation |Social Development |Emotion |Parent |sd_p_ders |abcd_socdev_p_emr01; soc_dev_fu_diff_emo_reg_p01 | | | | |- |Firearms (BRFSS) |Social Development |Firearm Storage |Parent |sd_p_fa |soc_dev_fu_firearms_p01; abcd_socdev_p_fa01 | | | | |- |Alabama Parenting Questionnaire |Social Development |Parenting |Parent |sd_p_apq |soc_dev_fu_alabama_p01; abcd_socdev_p_alabama01 | | | | |- |Personality Disposition |Social Development |Temperament/Personality |Parent |sd_p_pd |abcd_socdev_p_pdis01; soc_dev_fu_personality_p01 | | | | |- |Victimization |Social Development |Victimization |Parent |sd_p_vict |soc_dev_fu_victimize_p01; abcd_socdev_p_vic01 | | | | |} {| class="wikitable sortable mw-collapsible" |+ABCD Substudy - Linked Data Coding Information !Measure !Category !Subcategory !Source !Table Name !Table Name (NDA 4.0) !Subscale Information |- |Administrative Information |COVID-19 |Geocoded Data |Youth |cvd_l_geo_adm |N/A | |- |CDC Policy Surveilance |COVID-19 |Geocoded Data |CDC |cvd_l_geo_policy |N/A | |- |Johns Hopkins University COVID-19 Prevalence |COVID-19 |Geocoded Data |JHU |cvd_l_geo_jhu |N/A | |- |SafeGraph Social Distancing Metrics |COVID-19 |Geocoded Data |SafeGraph |cvd_l_geo_sg |N/A | |- |US Bureau of Labor Statistics (BLS) and Census Data |COVID-19 |Geocoded Data |BLS/Census |cvd_l_geo_bls |N/A | |} == Notes - Coding Information == ====== Physical Health ====== * Blood analyses run were taken from the online data dictionary. Blood (DNA) was listed as a distinct analysis in the PDFs of measures by wave but was not specifically mentioned in the data dictionary (hence why no table name was given). ** Oral fluids (DNA) were also not listed in the data dictionary. ====== Substance Use ====== *Caffeine use questions were not included under the substance use phone interview in the online data dictionary. == Variable, Subscale, and Total Score Information == The following table contains subscale information for measures used in the ABCD Study. {| class="wikitable sortable mw-collapsible" |+ABCD Core - Overview of Measures by Wave ''*Imaging data displayed separately below'' !Measure !Category !Subcategory !Informant !Subscale Information !Subscale Score Calculation !Total Scale Score Calculation |- |Occupation Survey |General Information |Demographics |Parent |'''No subscale information available'''. Different codes exist for a variety of occupation subcategories, which, when endorsed, prompt the informant to specify their job title. |N/A |N/A |- |PhenX Demographics Survey |General Information |Demographics |Parent |'''No subscale information available'''. Different variables exist for demographic information pertaining to the informant(s) (race, country of birth, years lived in US, marital status, schooling, income, occupation, access to goods/services/utilities, household, healthcare access), child (age, grade, adoption status, sex, gender, race, ethicity, nationality, household, religion, language), and other family members (race, nationality, income, occupation, relationship to informant and child). |N/A |N/A |- | | | | | | | |- |[[COVID-19]] Annual Form |Physical Health |COVID |Youth |'''No subscale information available'''. Scale assesses impact of COVID-19 on schooling, social distancing practice, interaction with other people, feelings of uncertainty, news consumption, and masking behavior. |N/A |N/A |- |Pain Questionnaire<ref>Luntamo, T., Sourander, A., Santalahti, P., Aromaa, M., & Helenius, H. (2012). Prevalence changes of pain, sleep problems and fatigue among 8-year-old children: years 1989, 1999, and 2005. ''Journal of pediatric psychology'', ''37''(3), 307–318.</ref> |Physical Health |Medical |Youth |'''No subscale information available'''. Scale assesses pain duration and impact over the past month, and separate variables exist for different places where the pain may have occurred. |N/A |N/A |- |Respiratory Functioning<ref>Gillman, M. W., & Blaisdell, C. J. (2018). Environmental influences on Child Health Outcomes, a Research Program of the National Institutes of Health. ''Current opinion in pediatrics'', ''30''(2), 260–262.</ref><ref>Asher, M. I., Keil, U., Anderson, H. R., Beasley, R., Crane, J., Martinez, F., Mitchell, E. A., Pearce, N., Sibbald, B., & Stewart, A. W. (1995). International Study of Asthma and Allergies in Childhood (ISAAC): rationale and methods. ''The European respiratory journal'', ''8''(3), 483–491.</ref> |Physical Health |Medical |Youth |'''No subscale information available'''. Different variables exist for presence and impact of wheezing/whistling, different triggers, asthma and pneumonia/bronchitis diagnosis information, and hospitalization. |N/A |N/A |- |Block Kids Food Screener - Youth<ref name=":1">Hunsberger, M., O’Malley, J., Block, T., & Norris, J. C. (2015). Relative validation of Block Kids Food Screener for dietary assessment in children and adolescents. ''Maternal & child nutrition'', ''11''(2), 260–270.</ref> |Physical Health |Nutrition |Youth |'''No subscale information available'''. Different variables exist for different foods, frequency of food consumption per day/week, and total grams consumed. |N/A |N/A |- |Sports and Activities Involvement Questionnaire (Reading & Music) - Youth |Physical Health |Physical Activity |Youth |'''Reading:''' sai_read_hrs_wk_y, sai_read_y, sai_read_enjoy_y '''Music:''' sai_lmusic_y, sai_lmusic_feel_y, sai_lmusic_hrs_day_y, sai_lmusic_studying_y |N/A |N/A |- |[[wikipedia:Youth_Risk_Behavior_Surveillance_System|Youth Risk Behavior Survey]] - Exercise |Physical Health |Physical Activity |Youth |'''No subscale information available'''. All items assess days spent engaging in physical activity per week. |N/A |N/A |- |Pubertal Development Scale and Menstrual Cycle Survey - Youth<ref name=":0">Petersen, A. C., Crockett, L., Richards, M., & Boxer, A. (1988). A self-report measure of pubertal status: Reliability, validity, and initial norms. ''Journal of youth and adolescence'', ''17''(2), 117–133.</ref> |Physical Health |Puberty |Youth |'''Male Pubertal Stage:''' pds_bdyhair_y, pds_m4_y, pds_m5_y '''Female Pubertal Stage:''' pds_bdyhair_y, pds_f4_2_y, pds_f5_y. '''Menstrual Cycle:''' menstrualcycle1_y, menstrualcycle2_y, menstrualcycle2_y_dk, menstrualcycle3_y, menstrualcycle4_y, menstrualcycle5_y, menstrualcycle6_y, menstrualcycle11_y, menstrualcycle7_y, menstrualcycle8_y, menstrualcycle9_y, menstrualcycle10_y, mentrualcycle11_y '''Other:''' pds_sex_y, pds_ht2_y, pds_skin2_y, pds_f6_y, pds_f6_y_dk |'''Male:''' Sum of pds_bdyhair_y, pds_m4_y, pds_m5_y. Prepubertal: 3; early pubertal: 4 or 5 (no responses of "3" to any item); midpubertal: 6-8 (no "4"s); late pubertal: 9-11; postpubertal (12). Subscale score variables are pds_y_ss_male_category and pds_y_ss_male_cat_2 (latter specifies sex assigned at birth). '''Female:''' Sum of pds_bdyhair_y and pds_f4_2_y, in addition to pds_f5_y (yes/no for menstruation). Prepubertal: 2 + no menstruation; early puberty: 3 + no menstruation; midpubertal: >3 + no menstruation; late puberty: ≤7 + menstruation; postpubertal: 8 + menstruation. Subscale score variables are pds_y_ss_female_category and pds_y_ss_female_category_2 (latter specifies sex assigned at birth). |N/A |- |[[wikipedia:Munich_Chronotype_Questionnaire|Munich ChronoType Questionnaire]] (sleep)<ref>Zavada, A., Gordijn, M. C., Beersma, D. G., Daan, S., & Roenneberg, T. (2005). Comparison of the Munich Chronotype Questionnaire with the Horne-Ostberg’s Morningness-Eveningness Score. ''Chronobiology international'', ''22''(2), 267–278.</ref> |Physical Health |Sleep |Youth |'''No subscale information available'''. Different variables exist for sleep times, sleep latency, awakenings, sleep inertia, alarm usage, sleep loss, social jetlag, timing of school/work, and sleep duration. Scale differentiates between sleep patterns on school/work days and free days. |N/A |N/A |- | | | | | | | |- |COVID-19 Annual Form |Physical Health |COVID |Parent |'''No subscale information available'''. Scale assesses impact of COVID-19 on employment, income, exposure to racism or discrimination, interpersonal relationships, stress, schooling (of child), access to goods and services (e.g., food, healthcare). Also assesses internet access, symptoms/diagnosis of COVID in child and family, impact of illness on child, conversations pertaining to COVID safety, vaccination, and hospitalization due to COVID. |N/A |N/A |- |Breast Feeding Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref> |Physical Health |Development |Parent |'''No subscale information available.''' Scale assesses prescription and non-prescription drug use during breastfeeding. |N/A |N/A |- |Developmental History Questionnaire<ref>Kessler, R. C., Avenevoli, S., Costello, E. J., Green, J. G., Gruber, M. J., Heeringa, S., Merikangas, K. R., Pennell, B. E., Sampson, N. A., & Zaslavsky, A. M. (2009). National comorbidity survey replication adolescent supplement (NCS-A): II. Overview and design. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 380–385.</ref><ref>Merikangas, K. R., Avenevoli, S., Costello, E. J., Koretz, D., & Kessler, R. C. (2009). National comorbidity survey replication adolescent supplement (NCS-A): I. Background and measures. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''48''(4), 367–379.</ref> |Physical Health |Development |Parent |'''No subscale information available.''' Scale assesses birthweight of child, age of biological parents, planning of pregnancy, twin status of child, drug use prior to knowledge of pregnancy, drug use after knowledge of pregnancy, experience with medical conditions during pregnancy, access to medical care during pregnancy, experience with birth complications, child illness in the first year of life, breastfeeding, timing of child motor development, timing of speech/linguistic development, and child bed wetting. |N/A |N/A |- |Medical History Questionnaire<ref>Todd, R. D., Joyner, C. A., Heath, A. C., Neuman, R. J., & Reich, W. (2003). Reliability and stability of a semistructured DSM-IV interview designed for family studies. ''Journal of the American Academy of Child and Adolescent Psychiatry'', ''42''(12), 1460–1468.</ref> |Physical Health |Medical |Parent |'''No subscale information available.''' Asks about child's experience with illness, disabilityt, injury, hospitalization/urgent care/primary care, and surgery. |N/A |N/A |- |Ohio State [[wikipedia:Traumatic_brain_injury|TBI]] Screen - Short<ref>Bogner, J. A., Whiteneck, G. G., MacDonald, J., Juengst, S. B., Brown, A. W., Philippus, A. M., Marwitz, J. H., Lengenfelder, J., Mellick, D., Arenth, P., & Corrigan, J. D. (2017). Test-Retest Reliability of Traumatic Brain Injury Outcome Measures: A Traumatic Brain Injury Model Systems Study. ''The Journal of head trauma rehabilitation'', ''32''(5), E1–E16. </ref> |Physical Health |Medical |Parent |'''No subscale information available.''' Scale assesses child's history of head injury, including diagnosis, loss of consciousness, age, nature of injury, amnesia, frequency and timing of injuries, and lasting impact of injury. |N/A |N/A |- |PhenX Medications Survey (Medications Inventory) |Physical Health |Medical |Parent |'''No subscale information available.''' Scale measures child's medication usage within the last two weeks and the last year. Up to 15 prescription medications and 15 OTC medications can be inputted. Variables exist for medication name, dosage, release (e.g., XR), and frequency of consumption. Additional items assess caffeine and CBD use. |N/A |N/A |- |Child Nutrition Assessment<ref>Morris, M. C., Tangney, C. C., Wang, Y., Sacks, F. M., Bennett, D. A., & Aggarwal, N. T. (2015). MIND diet associated with reduced incidence of Alzheimer’s disease. ''Alzheimer’s & dementia : the journal of the Alzheimer’s Association'', ''11''(9), 1007–1014.</ref> |Physical Health |Nutrition |Parent |'''No subscale information available.''' All items contribute to one total variable (diet score, ''cna_p_ss_sum''). Scale assesses the child's typical consumption of various foods and food groups over the past year, and if any prenatal vitamins or folic acid supplements were taken by the biological mother prior to or during pregnancy. |N/A |N/A |- |Block Kids Food Screener - Parent<ref name=":1" /> |Physical Health |Nutrition |Parent |'''No subscale information available'''. Different variables exist for different foods, frequency of food consumption per day/week, and total grams consumed. |N/A |N/A |- |International Physical Activity Questionnaire<ref>Booth M. (2000). Assessment of physical activity: an international perspective. ''Research quarterly for exercise and sport'', ''71''(2 Suppl), S114–S120.</ref> |Physical Health |Physical Activity |Parent |'''No subscale information available'''. Scale assesses time spent walking, engaging in vigorous/moderate physical activity, and walking over the past week. |N/A |N/A |- |Sports and Activities Involvement Questionnaire - Parent<ref>Huppertz, C., Bartels, M., de Zeeuw, E. L., van Beijsterveldt, C., Hudziak, J. J., Willemsen, G., Boomsma, D. I., & de Geus, E. (2016). Individual Differences in Exercise Behavior: Stability and Change in Genetic and Environmental Determinants From Age 7 to 18. ''Behavior genetics'', ''46''(5), 665–679.</ref> |Physical Health |Physical Activity |Parent |'''No subscale information available'''. Scale assesses child participation in various sports and activities continuously for 4 months or more over the past year. |N/A |N/A |- |Pubertal Development Scale and Menstrual Cycle Survey - Parent<ref name=":0" /> |Physical Health |Puberty |Parent |'''Male Pubertal Stage:''' pds_2_p, pds_m5_p, pds_m4_p '''Female Pubertal Stage:''' pds_2_p, pds_f4_p, pds_f5b_p. '''Menstrual Cycle:''' menstrualcycle1_p, menstrualcycle2_p, menstrualcycle2_p_dk, menstrualcycle3_p, menstrualcycle4_p, menstrualcycle5_p, menstrualcycle6_p '''Other:''' pubertal_sex_p, pds_1_p, pds_3_p, pds_f6_p, pds_f6_p_dk |'''Male:''' Sum of pds_2_p, pds_m5_p, and pds_m4_p. Prepubertal: 3; early pubertal: 4 or 5 (no responses of "3" to any item); midpubertal: 6-8 (no "4"s); late pubertal: 9-11; postpubertal (12). Subscale score variables are pds_p_ss_male_category and pds_p_ss_male_category_2 (latter specifies sex assigned at birth). '''Female:''' Sum of pds_2_p and pds_f4_p, in addition to pds_f5b_p (yes/no for menstruation). Prepubertal: 2 + no menstruation; early puberty: 3 + no menstruation; midpubertal: >3 + no menstruation; late puberty: ≤7 + menstruation; postpubertal: 8 + menstruation. Subscale score variables are pds_p_ss_female_category and pds_p_ss_female_category_2 (latter specifies sex assigned at birth). |N/A |- |Sleep Disturbances Scale for Children<ref>Bruni, O., Ottaviano, S., Guidetti, V., Romoli, M., Innocenzi, M., Cortesi, F., & Giannotti, F. (1996). The Sleep Disturbance Scale for Children (SDSC). Construction and validation of an instrument to evaluate sleep disturbances in childhood and adolescence. ''Journal of sleep research'', ''5''(4), 251–261.</ref><ref>Ferreira, V. R., Carvalho, L. B., Ruotolo, F., de Morais, J. F., Prado, L. B., & Prado, G. F. (2009). Sleep disturbance scale for children: translation, cultural adaptation, and validation. ''Sleep medicine'', ''10''(4), 457–463.</ref> |Physical Health |Sleep |Parent |'''Disorders of Initiating and Maintaining Sleep (DIMS):''' sleepdisturb1_p, sleepdisturb2_p, sleepdisturb3_p, sleepdisturb4_p, sleepdisturb5_p, sleepdisturb10_p, sleepdisturb11_p '''Sleep Breathing Disorders (SBD):''' sleepdisturb13_p, sleepdisturb14_p, sleepdisturb15_p '''Disorder of Arousal (DA):''' sleepdisturb17_p, sleepdisturb20_p, sleepdisturb21_p '''Sleep-Wake Transition Disorders (SWTD):''' sleepdisturb6_p, sleepdisturb7_p, sleepdisturb8_p, sleepdisturb12_p, sleepdisturb18_p, sleepdisturb19_p '''Disorders of Excessive Somnolence (DOES):''' sleepdisturb22_p, sleepdisturb23_p, sleepdisturb24_p, sleepdisturb25_p, sleepdisturb26_p '''Sleep Hyperhydrosis (SHY):''' sleepdisturb9_p, sleepdisturb16_p |All subscale scores are sums. '''DIMS:''' sds_p_ss_dims; '''SBD:''' sds_p_ss_sbd; '''DA:''' sds_p_ss_da; '''SWTD:''' sds_p_ss_swtd; '''DOES:''' sds_p_ss_does; '''SHY:''' sds_p_ss_shy. |Total score is equal to the sum of all subscale scores. '''sds_p_ss_total''' |- | | | | | | | |- |Gender Identity (Youth)<ref name=":16">Potter, A., Dube, S., Allgaier, N., Loso, H., Ivanova, M., Barrios, L. C., Bookheimer, S., Chaarani, B., Dumas, J., Feldstein-Ewing, S., Freedman, E. G., Garavan, H., Hoffman, E., McGlade, E., Robin, L., & Johns, M. M. (2021). Early adolescent gender diversity and mental health in the Adolescent Brain Cognitive Development study. ''Journal of child psychology and psychiatry, and allied disciplines'', ''62''(2), 171–179.</ref><ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Reed, E., Salazar, M., Behar, A. I., Agah, N., Silverman, J. G., Minnis, A. M., Rusch, M., & Raj, A. (2019). Cyber Sexual Harassment: Prevalence and association with substance use, poor mental health, and STI history among sexually active adolescent girls. ''Journal of adolescence'', ''75'', 53–62.</ref> |Gender & Sexuality |Gender |Youth |'''Male Gender:''' gish_m1_y, gish_m2_y, gish_m3_y, gish_m4_y. '''Female Gender:''' gish_f1_y, gish_f2_y, gish_f3_y, gish_f4_y. '''Other:''' y_gish2_desc_fem, y_gish2_desc_fem_self, y_gish2_desc_male, y_gish2_desc_male_self, kbi_sex_assigned_at_birth, kbi_gender, kbi_y_trans_id, kbi_y_trans_prob |Both subscale scores are averages. '''Male Gender:''' gish_y_ss_m_avg '''Female Gender:''' gish_y_ss_f_avg |N/A |- |Sexual Behavior/Health<ref>Potter, A. S., Dube, S. L., Barrios, L. C., Bookheimer, S., Espinoza, A., Feldstein Ewing, S. W., Freedman, E. G., Hoffman, E. A., Ivanova, M., Jefferys, H., McGlade, E. C., Tapert, S. F., & Johns, M. M. (2022). Measurement of gender and sexuality in the Adolescent Brain Cognitive Development (ABCD) study. ''Developmental cognitive neuroscience'', ''53'', 101057. </ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref><ref>Windle, M., Grunbaum, J. A., Elliott, M., Tortolero, S. R., Berry, S., Gilliland, J., Kanouse, D. E., Parcel, G. S., Wallander, J., Kelder, S., Collins, J., Kolbe, L., & Schuster, M. (2004). Healthy passages. A multilevel, multimethod longitudinal study of adolescent health. ''American journal of preventive medicine'', ''27''(2), 164–172.</ref> |Gender & Sexuality |Sexuality |Youth |'''No subscale information available.''' Different variables exist for a range of romantic and sexual experiences a child may have had, sexual orientation, interpersonal effects of this, and substance use. |N/A |N/A |- | | | | | | | |- |Gender Identity (Parent)<ref name=":16" /><ref>Johnson, L. L., Bradley, S. J., Birkenfeld-Adams, A. S., Kuksis, M. A., Maing, D. M., Mitchell, J. N., & Zucker, K. J. (2004). A parent-report gender identity questionnaire for children. ''Archives of sexual behavior'', ''33''(2), 105–116.</ref><ref>Elizabeth, P. H., & Green, R. (1984). Childhood sex-role behaviors: similarities and differences in twins. ''Acta geneticae medicae et gemellologiae'', ''33''(2), 173–179.</ref> |Gender & Sexuality |Gender |Parent |'''Male Expression:''' gish_m1_p, gish_m2_p, gish_m3_p, gish_m4_p, gish_m5_p, gish_m6_p, gish_m7_p, gish_m8_p, gish_m9_p, gish_m10_p, gish_m11_p. '''Female Expression:''' gish_f1_p, gish_f2_p, gish_f3_p, gish_f4_p, gish_f5_p, gish_f6_p, gish_f7_p, gish_f8_p, gish_f9_p, gish_f10_p, gish_f11_p. '''Male Dysphoria:''' gish_m12_p, gish_m13_p, gish_m14_p. '''Female Dysphoria:''' gish_f12_p, gish_f13_p, gish_f14_p. '''Male GIQ:''' gish_m1_p - gish_m14_p '''Female GIQ:''' gish_f1_p - gish_f14_p '''Other:''' p_gish_desc_male, p_gish_desc_male_self, p_gish_he_wish, p_gish_desc_fem, p_gish_desc_fem_self, p_gish_she_wish, demo_sex_v2, demo_gender_id_v2, demo_gender_id_v2_l, kbi_p_c_trans, kbi_p_c_trans_l, kbi_p_c_trans_problems, kbi_p_c_trans_problems_l |All subscale scores are averages. '''Male Expression:''' gish_p_ss_m_exp_avg; '''Female Expression:''' gish_p_ss_f_exp_avg; '''Male Dysphoria:''' gish_p_ss_m_dys_avg; '''Female Dysphoria:''' gish_p_ss_f_dys_avg; '''Male GIQ:''' gish_p_ss_m_avg, '''Feamle GIQ:''' gish_p_ss_f_avg. |N/A |- |Sexual Behavior/Health<ref name=":16" /><ref>Wylie, S. A., Corliss, H. L., Boulanger, V., Prokop, L. A., & Austin, S. B. (2010). Socially assigned gender nonconformity: A brief measure for use in surveillance and investigation of health disparities. ''Sex roles'', ''63''(3-4), 264–276.</ref><ref>Sales, J. M., Milhausen, R. R., Wingood, G. M., Diclemente, R. J., Salazar, L. F., & Crosby, R. A. (2008). Validation of a Parent-Adolescent Communication Scale for use in STD/HIV prevention interventions. ''Health education & behavior : the official publication of the Society for Public Health Education'', ''35''(3), 332–345.</ref> |Gender & Sexuality |Sexuality |Parent |'''No subscale information available.''' Scale assesses caregiver's perception of their child's sexuality, interpersonal effects of this, and if the caregiver has talked with their child about sex |N/A |N/A |- | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |'''No subscale information available.''' |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Youth |'''No subscale information available.''' |N/A |N/A |- |Brief Problem Monitor Scale<ref>Achenbach, T. M. (2009). The Achenbach System of Empirically Based Assessment (ASEBA): Development, Findings, Theory, and Applications. Burlington, VT: University of Vermont Research Center for Children, Youth, & Families.</ref> |Mental Health |Broad Psychopathology |Youth |'''Attention:''' bpm_1_y, bpm_3_y, bpm_4_y, bpm_5_y, bpm_10_y, bpm_14_y '''Externalizing:''' bpm_2_y, bpm_6_y, bpm_7_y, bpm_8_y, bpm_15_y, bpm_16_y, bpm_17_y '''Internalizing:''' bpm_9_y, bpm_11_y, bpm_12_y, bpm_13_y, bpm_18_y, bpm_19_y |Raw total, mean, and sex-normed T scores exist for all subscales. '''Attention:''' bpm_y_scr_attention_r, bpm_y_scr_attention_t, bpm_y_ss_attention_mean; '''Externalizing:''' bpm_y_scr_external_r, bpm_y_scr_external_t, bpm_y_ss_external_mean; '''Internalizing:''' bpm_y_scr_internal_r, bpm_y_scr_internal_t, bpm_y_ss_internal_mean |Raw total, mean, and sex-normed T scores exist for the BPMS. '''bpm_y_scr_totalprob_r, bpm_y_scr_totalprob_t, bpm_y_ss_totalprob_mean''' |- |KSADS - Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Youth |'''No subscale information available.''' Contains variables for all possible KSADS discrete symptoms and diagnostic labels. |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Youth |'''No subscale information available.''' |N/A |N/A |- |Emotion Regulation Questionnaire<ref>Gross, J.J., & John, O.P. (2003). Individual differences in two emotion regulation processes: Implications for affect, relationships, and well-being. Journal of Personality and Social Psychology, 85, 348-362. </ref><ref>Garnefski, N., Rieffe, C., Jellesma, F., Terwogt, M. M., & Kraaij, V. (2007). Cognitive emotion regulation strategies and emotional problems in 9–11-year-old children: The development of an instrument. European Child & Adolescent Psychiatry, 16, 1–9.</ref><ref>Gullone, E., & Taffe, J. (2012). The Emotion Regulation Questionnaire for Children and Adolescents (ERQ-CA): a psychometric evaluation. Psychological assessment, ''24''(2), 409–417.</ref> |Mental Health |Emotion |Youth |'''Cognitive Reappraisal:''' erq_feelings_think, erq_feelings_happy, erq_feelings_less_bad '''Expressive Suppression:''' erq_feelings_self, erq_feelings_control, erq_feelings_hide |Both subscale scores are prorated sums. '''Cognitive Reappraisal:''' erq_ss_reappraisal_pr; '''Expressive Suppression:''' erq_ss_suppress_pr |N/A |- |NIH Toolbox Positive Affect Items<ref>Salsman, J. M., Butt, Z., Pilkonis, P. A., Cyranowski, J. M., Zill, N., Hendrie, H. C., Kupst, M. J., Kelly, M. A. R., Bode, R. K., Choi, S. W., Lai, J.-S. ., Griffith, J. W., Stoney, C. M., Brouwers, P., Knox, S. S., & Cella, D. (2013). Emotion assessment using the NIH Toolbox. ''Neurology'', ''80''(Issue 11, Supplement 3), S76–S86. <nowiki>https://doi.org/10.1212/wnl.0b013e3182872e11</nowiki> </ref><ref>{{Cite web|url=https://nihtoolbox.org/test/positive-affect/|title=Positive Affect|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref>Gershon, R.C., Wagster, M.V., et al., 2013. NIH toolbox for assessment of neurological and behavioral function. Neurology 80 (11 Suppl. 3), S2–6.</ref> |Mental Health |Emotion |Youth |'''No subscale information available.''' Scale assesses child's self-appraisal of positive affect. |N/A |Total score is a sum. '''poa_y_ss_sum''' |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Youth |'''No subscale information available.''' |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Youth |'''No subscale information available.''' |N/A |N/A |- |[[OToPS/Measures/7 Up 7 Down Inventory|7-Up Mania Items]]<ref name=":9">Youngstrom, E. A., Murray, G., Johnson, S. L., & Findling, R. L. (2013). The 7 Up 7 Down Inventory: A 14-item measure of manic and depressive tendencies carved from the General Behavior Inventory. ''Psychological Assessment'', ''25''(4), 1377–1383. <nowiki>https://doi.org/10.1037/a0033975</nowiki> </ref> |Mental Health |Mood |Youth |'''No subscale information available.''' Scale assesses symptoms of mania. |N/A |Total score is a sum. '''sup_y_ss_sum''' |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |'''No subscale information available.''' |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |'''No subscale information available.''' |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Youth<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Youth |'''No subscale information available.''' |N/A |N/A |- |Peer Experiences Questionnaire<ref>De Los Reyes, A. & Prinstein, M. J. (2004). Applying depression-distortion hypotheses to the assessment of peer victimization in adolescents. Journal of Clinical Child and Adolescent Psychology, 33, 325-335.</ref><ref>Prinstein, M. J., Boergers, J., & Vernberg, E. M. (2001). Overt and relational aggression in adolescents: Social-psychological functioning of aggressors and victims. Journal of Clinical Child Psychology, 30, 477-489.</ref> |Mental Health |Peers |Youth |'''Overt Aggression:''' peq_chase_perp, peq_threat_perp, peq_hit_perp '''Overt Victimization:''' peq_chase_vic, peq_threat_vic, peq_hit_vic '''Relational Aggression:''' peq_left_out_perp, peq_invite_perp, peq_exclude_perp '''Relational Victimization:''' peq_left_out_vic, peq_invite_vic, peq_exclude_vic '''Reputational Aggression:''' peq_rumor_perp, peq_gossip_perp, peq_loser_perp '''Reputational Victimization:''' peq_rumor_vic, peq_gossip_vic, peq_loser_vic |All subscale scores are sums. '''Overt Aggression:''' peq_ss_overt_aggression; '''Overt Victimization:''' peq_ss_overt_victim; '''Relational Aggression:''' peq_ss_relational_aggs; '''Relational Victimization:''' peq_ss_relational_victim; '''Reputational Aggression:''' peq_ss_reputation_aggs; '''Reputational Victimization:''' peq_ss_reputation_victim |N/A |- |Youth Resilience Scale<ref>Ungar, M., & Liebenberg, L. (2009). Cross-cultural consultation leading to the development of a valid measure of youth resilience: The International Resilience Project. ''Studia psychologica'', ''51''(2-3), 259-268.</ref> |Mental Health |Peers |Youth |'''No subscale information available.''' Scale assesses how many (close) friends a child has that are girls, boys, or nonbinary |N/A |N/A |- |Prodromal Psychosis Scale<ref>Karcher NR, Barch DM, Avenevoli S, Savill M, Huber RS, Simon TJ, Leckliter IN, Sher KJ, Loewy RL. Assessment of the Prodromal Questionnaire-Brief Child Version for Measurement of Self-reported Psychotic like Experiences in Childhood. JAMA Psychiatry. 2018 Aug 1;75(8):853-861.</ref><ref>Loewy, R.L., Bearden, C.E., et al., 2005. The prodromal questionnaire (PQ): preliminary validation of a self-report screening measure for prodromal and psychotic syndromes. Schizophr. Res. 79 (1), 117–125.</ref><ref>Ising, H.K., Veling, W., et al., 2012. The validity of the 16-item version of the Prodromal Questionnaire (PQ-16) to screen for ultra high risk of developing psychosis in the general help-seeking population. Schizophr. Bull. 38 (6), 1288–1296.</ref><ref>Therman, S., Lindgren, M., et al., 2014. Predicting psychosis and psychiatric hospital care among adolescent psychiatric patients with the Prodromal Questionnaire. Schizophr. Res. 158 (1–3), 7–10.</ref> |Mental Health |Psychosis |Youth |'''Experiences of Psychosis:''' prodromal_1_y ''through'' prodromal_21_y; '''Did this Experience of Psychosis Bother You:''' pps_1_bother_yn ''through'' pps_21_bother_yn. |All subscales are sums. For the "Did the Experience of Psychosis Bother You" subscale, sum variables for both "yes" and "no" responses exist and the subscale score is equal to the total number of "yes" or "no" responses. '''Experiences of Psychosis:''' pps_y_ss_number; '''Did this Experience of Psychosis Bother You:''' pps_y_ss_bother_sum, pps_y_ss_bother_n_1 |The total score is a severity score, calculated by adding up responses to all questions asking "how much did this experience [of psychosis] bother you?" (scale of 1-5; ), plus the "Did this Experience of Psychosis Bother You?" subscale score. '''pps_y_ss_severity_score''' |- |[[wikipedia:Kiddie_Schedule_for_Affective_Disorders_and_Schizophrenia|Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)]] Background Items Survey - Youth<ref name=":6">KAUFMAN, J., BIRMAHER, B., BRENT, D., RAO, U., FLYNN, C., MORECI, P., WILLIAMSON, D., & RYAN, N. (1997). Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version (K-SADS-PL): Initial Reliability and Validity Data. ''Journal of the American Academy of Child & Adolescent Psychiatry'', ''36''(7), 980–988. <nowiki>https://doi.org/10.1097/00004583-199707000-00021</nowiki></ref><ref name=":7">Kaufman, J., Birmaher, B., Axelson, D., Perepletchikova, F., Brent, D., & Ryan, N. (2016). ''K-SADS-PL DSM-5''. <nowiki>https://pediatricbipolar.pitt.edu/sites/default/files/assets/Clinical%20tools/KSADS/KSADS_DSM_5_SCREEN_Final.pdf</nowiki> </ref><ref name=":2">Townsend, L, Kobak, K., Kearney, C., Milham, M., Andreotti, C., Escalera, J., Alexander, L., Gill, M.K., Birmaher, B., Sylvester, R., Rice, D., Deep, A., Kaufman, J. (2020).  Development of Three Web-Based Computerized Versions of the Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS-COMP) Child Psychiatric Diagnostic Interview: Preliminary Validity Data. Journal of the American Academy of Child and Adolescent Psychiatry, Feb;59(2):309-325. doi:10.1016/j.jaac. PMID: 31108163.</ref><ref name=":3">Kaufman, J., Kobak, K., Birmaher, B., & de Lacy, N. (2021). KSADS-COMP Perspectives on Child Psychiatric Diagnostic Assessment and Treatment Planning. Journal of the American Academy of Child and Adolescent Psychiatry, ''60''(5), 540–542.</ref> |Mental Health |Psychosocial |Youth |'''No subscale information available.''' Assesses gender identity and sexuality, and issues at home, with friends, and at school. |N/A |N/A |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems)<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Youth |'''No subscale information available.''' |N/A |N/A |- |Cyberbullying Questionnaire<ref>Stewart RW, Drescher CF, Maack DJ, Ebesutani C, Young J. The Development and Psychometric Investigation of the Cyberbullying Scale. J Interpers Violence. 2014 Aug;29(12):2218-2238. doi: 10.1177/0886260513517552. Epub 2014 Jan 14. PMID: 24424252.</ref> |Mental Health |Social |Youth |'''No subscale information available.''' Scale assesses whether a child has been a victim or perpatraitor of cyberbullying within the past year, how often this behavior occurred, and the power dynamics present in this relationship. |N/A |N/A |- |Life Events Scale - Youth<ref name=":10">Tiet, Q.Q., Bird, H.R., et al., 2001. Relationship between specific adverse life events and psychiatric disorders. J. Abnorm. Child Psychol. 29 (2), 153–164.</ref><ref name=":15">Grant, K.E., Compas, B.E., et al., 2004. Stressors and child and adolescent psychopathology: measurement issues and prospective effects. J. Clin. Child Adolesc. Psychol. 33 (2), 412–425.</ref> |Mental Health |Stress |Youth |'''Events:''' ple_died_y, ple_injured_y, ple_crime_y, ple_friend_y, ple_friend_injur_y, ple_financial_y, ple_sud_y, ple_ill_y, ple_injur_y, ple_argue_y, ple_job_y, ple_away_y, ple_arrest_y, ple_friend_died_y, ple_mh_y, ple_sib_y, ple_victim_y, ple_separ_y, ple_law_y, ple_school_y, ple_move_y, ple_jail_y, ple_step_y, ple_new_job_y, ple_new_sib_y, ple_foster_care_y, ple_hit_y, ple_homeless_y, ple_hospitalized_y, ple_lockdown_y, ple_shot_y, ple_suicide_y, ple_deported_y '''Mostly Good/Mostly Bad Events:''' ''A response of "1" to the following items indicates a mostly good experience, while a response of "2" indicates a mostly bad experience.'' ple_died_fu_y, ple_injured_fu_y, ple_crime_fu_y, ple_friend_fu_y, ple_friend_injur_fu_y, ple_financial_fu_y, ple_sud_fu_y, ple_ill_fu_y, ple_argue_fu_y, ple_job_fu_y, ple_away_fu_y, ple_arrest_fu_y, ple_friend_died_fu_y, ple_mh_fu_y, ple_sib_fu_y, ple_victim_fu_y, ple_separ_fu_y, ple_law_fu_y, ple_school_fu_y, ple_move_fu_y, ple_jail_fu_y, ple_step_fu_y, ple_new_job_fu_y, ple_new_sib_fu_y, ple_hit_fu_y, ple_homeless_fu_y, ple_hospitalized_fu_y, ple_lockdown_fu_y, ple_shot_fu_y, ple_deported_fu_y, ple_foster_care_fu_y, ple_injur_fu_y '''How Much Affected:''' ple_died_fu2_y, ple_injured_fu2_y, ple_crime_fu2_y, ple_friend_fu2_y, ple_friend_injur_fu2_y, ple_financial_fu2_y, ple_sud_fu2_y, ple_ill_fu2_y, ple_injur_fu2_y, ple_argue_fu2_y, ple_job_fu2_y, ple_away_fu2_y, ple_arrest_fu2_y, ple_friend_died_fu2_y, ple_mh_fu2_y, ple_sib_fu2_y, ple_victim_fu2_y, ple_separ_fu2_y, ple_law_fu2_y, ple_school_fu2_y, ple_move_fu2_y, ple_jail_fu2_y, ple_step_fu2_y, ple_new_job_fu2_y, ple_new_sib_fu2_y, ple_hit_fu2_y, ple_homeless_fu2_y, ple_hospitalized_fu2_y, ple_lockdown_fu2_y, ple_shot_fu2_y, ple_suicide_fu2_y, ple_deported_fu2_y, ple_foster_care_fu2_y |All subscale scores are sums. Variables describing mean impact of "mostly good" and "mostly bad" events also exist. '''Total Number of Events:''' ple_y_ss_total_number; '''Total Number of Good Events:''' ple_y_ss_total_good; '''Total Number of Bad Events:''' ple_y_ss_total_bad; '''How Much Affected:''' ple_y_ss_affect_sum, ple_y_ss_affected_good_sum, ple_y_ss_affected_good_mean (only includes experiences that the child said were "mostly good"), ple_y_ss_affected_bad_sum, ple_y_ss_affected_bad_mean, (only includes experiences that the child said were "mostly bad"). |N/A |- |[[Behavioral Inhibition and Behavioral Activation System (BIS/BAS) Scales|PhenX Behavioral Inhibition/Behavioral Approach System (BIS/BAS) Scales]]<ref>Carver, C. & White, T. (1994). Behavioral Inhibition, Behavioral Activation, and affective response to impending reward and punishment: The BIS/BAS Scales. ''Journal of Personality and Social Psychology'', 67(2), 319-333.</ref><ref>Pagliaccio D, Luking KR, Anokhin AP, Gotlib IH, Hayden EP, Olino TM, Peng CZ, Hajcak G, Barch DM. Revising the BIS/BAS Scale to study development: Measurement invariance and normative effects of age and sex from childhood through adulthood. Psychol Assess. 2016 Apr;28(4):429-42. doi: 10.1037/pas0000186. Epub 2015 Aug 24. PMID: 26302106; PMCID: PMC4766059.</ref> |Mental Health |Temperament/Personality |Youth |'''BAS Drive:''' bisbas13_y, bisbas14_y, bisbas15_y, bisbas16_y '''BAS Fun Seeking:''' bisbas17_y, bisbas18_y, bisbas19_y, bisbas20_y '''BAS Reward Responsiveness:''' bisbas8_y, bisbas9_y, bisbas10_y, bisbas11_y, bisbas12_y '''BIS:''' bisbas1_y, bisbas2_y, bisbas3_y, bisbas4_y, bisbas5_y, bisbas6_y, bisbas7_y |All subscale scores are sums. Both modified and unmodified variables are included where applicable (see notes for more information) '''BAS Drive:''' bis_y_ss_basm_drive''',''' bis_y_ss_bas_drive; '''BAS Fun Seeking:''' bis_y_ss_bas_fs; '''BAS Reward Responsiveness:''' bis_y_ss_basm_rr, bis_y_ss_bas_rr; '''BIS:''' bis_y_ss_bis_sum, bis_y_ss_bism_sum |N/A |- |Urgency, Premeditation, Perseverance, Sensation Seeking, Positive Urgency, Impulsive Behavior (UPPS-P) for Children - Short Form (ABCD Version)<ref>Whiteside, S. P., & Lynam, D. R. (2001). The Five Factor Model and impulsivity: using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669-689. doi: 10.1016/S0191-8869(00)00064-7</ref><ref>Cyders, M. A., Smith, G. T., Spillane, N. S., Fischer, S., Annus, A. M., & Peterson, C. (2007). Integration of impulsivity and positive mood to predict risky behavior: Development and validation of a measure of positive urgency. ''Psychological Assessment, 19''(1), 107–118. <nowiki>https://doi.org/10.1037/1040-3590.19.1.107</nowiki></ref> |Mental Health |Temperament/Personality |Youth |'''Negative Urgency:''' upps7_y, upps11_y, upps17_y, upps20_y '''Lack of Planning:''' upps6_y, upps16_y, upps23_y, upps28_y '''Sensation Seeking:''' upps12_y, upps18_y, upps21_y, upps27_y '''Positive Urgency:''' upps35_y, upps36_y, upps37_y, upps39_y '''Lack of Perseverance:''' upps15_y, upps19_y, upps22_y, upps24_y |All subscale scores are sums. '''Negative Urgency:''' upps_y_ss_negative_urgency; '''Lack of Planning:''' upps_y_ss_lack_of_planning; '''Sensation Seeking:''' upps_y_ss_sensation_seeking; '''Positive Urgency:''' upps_y_ss_positive_urgency; '''Lack of Perseverance:''' upps_y_ss_lack_of_perseverance |N/A |- | | | | | | | |- |KSADS Diagnostic Interview for DSM-5 (Agoraphobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Generalized Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Obsessive Compulsive Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Panic Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Separation Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Social Anxiety Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Specific Phobia Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Anxiety |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Autism Spectrum Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Autism Spectrum |Parent | | | |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Behavior Checklist]]<ref name=":11">Achenbach, T. M., & Rescorla, L. A. (2003). Manual for the ASEBA adult forms & profiles. Research Center for Children, Youth, & Families, University of Vermont, Burlington, VT, USA.</ref> |Mental Health |Broad Psychopathology |Parent | | | |- |[[wikipedia:Achenbach_System_of_Empirically_Based_Assessment|ASEBA Adult Self-Report]] (psychopathology)<ref name=":11" /> |Mental Health |Broad Psychopathology |Parent | | | |- |[[wikipedia:Child_Behavior_Checklist|Child Behavior Checklist]]<ref>Achenbach TM, Rescorla LA. ''Manual for the ASEBA school-age forms & profiles: an integrated system of mult-informant assessment.'' Burlington: University of Vermont, Research Center for Children, Youth & Families; 2001.</ref> |Mental Health |Broad Psychopathology |Parent | | | |- |Family History Assessment Survey<ref>Brown SA, Brumback T, Tomlinson K, Cummins K, Thompson WK, Nagel BJ, De Bellis MD, Hooper SR, Clark DB, Chung T, Hasler BP, Colrain IM, Baker FC, Prouty D, Pfefferbaum A, Sullivan EV, Pohl KM, Rohlfing T, Nichols BN, Chu W, Tapert SF. The National Consortium on Alcohol and NeuroDevelopment in Adolescence (NCANDA): A Multisite Study of Adolescent Development and Substance Use. J Stud Alcohol Drugs. 2015 Nov;76(6):895-908. doi: 10.15288/jsad.2015.76.895. PMID: 26562597; PMCID: PMC4712659.</ref><ref>Rice JP, Reich T, Bucholz KK, Neuman RJ, Fishman R, Rochberg N, Hesselbrock VM, Nurnberger JI Jr, Schuckit MA, Begleiter H. Comparison of direct interview and family history diagnoses of alcohol dependence. Alcohol Clin Exp Res. 1995 Aug;19(4):1018-23. doi: 10.1111/j.1530-0277.1995.tb00983.x. PMID: 7485811.</ref> |Mental Health |Broad Psychopathology |Parent | | | |- |KSADS Symptoms & Diagnoses |Mental Health |Broad Psychopathology |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Eating Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Eating |Parent | | | |- |Difficulty in Emotion Regulation Scale<ref>Bardeen, J. R., Fergus, T. A., Hannan, S. M., & Orcutt, H. K. (2016). Addressing psychometric limitations of the Difficulties in Emotion Regulation Scale through item modification. Journal of Personality Assessment.</ref><ref>Bunford, N., Dawson, A. E., Evans, S. W., Ray, A. R., Langberg, J. M., Owens, J. S., DuPaul, G. J., & Allan, D. M. (2020). The Difficulties in Emotion Regulation Scale-Parent Report: A Psychometric Investigation Examining Adolescents With and Without ADHD. Assessment, 27(5), 921–940.</ref> |Mental Health |Emotion |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (ADHD) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Conduct Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Oppositional Defiant Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Externalizing |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Homicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Suicidality) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Harm |Parent | | | |- |[[OToPS/Measures/7 Up 7 Down Inventory|General Behavior Inventory - Mania]]<ref name=":9" /> |Mental Health |Mood |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Bipolar and Related Disorders) - Parent <ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Depressive Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Disruptive Mood Dysregulation Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Mood |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Psychotic Disorders) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosis |Parent | | | |- |KSADS Background Items Survey - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Psychosocial |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Sleep Problems) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Sleep |Parent | | | |- |Short Social Responsiveness Scale<ref>Aldridge, F. J., Gibbs, V. M., Schmidhofer, K., & Williams, M. (2012). Investigating the clinical usefulness of the Social Responsiveness Scale (SRS) in a tertiary level, autism spectrum disorder specific assessment clinic. ''Journal of autism and developmental disorders'', ''42''(2), 294–300.</ref><ref>Constantino J. N. (2011). The quantitative nature of autistic social impairment. ''Pediatric research'', ''69''(5 Pt 2), 55R–62R.</ref><ref>Constantino, J. N., Przybeck, T., Friesen, D., & Todd, R. D. (2000). Reciprocal social behavior in children with and without pervasive developmental disorders. ''Journal of developmental and behavioral pediatrics : JDBP'', ''21''(1), 2–11.</ref><ref>Constantino, J. N., & Todd, R. D. (2000). Genetic structure of reciprocal social behavior. ''The American journal of psychiatry'', ''157''(12), 2043–2045.</ref><ref>Constantino, J. N., & Todd, R. D. (2003). Autistic traits in the general population: a twin study. ''Archives of general psychiatry'', ''60''(5), 524–530.</ref><ref>Constantino, J. N., Gruber, C. P., Davis, S., Hayes, S., Passanante, N., & Przybeck, T. (2004). The factor structure of autistic traits. ''Journal of child psychology and psychiatry, and allied disciplines'', ''45''(4), 719–726.</ref><ref>Hus, V., Bishop, S., Gotham, K., Huerta, M., & Lord, C. (2013). Factors influencing scores on the social responsiveness scale. ''Journal of child psychology and psychiatry, and allied disciplines'', ''54''(2), 216–224.</ref><ref>Kaat, A. J., & Farmer, C. (2017). Commentary: Lingering questions about the Social Responsiveness Scale short form. A commentary on Sturm et al. (2017). ''Journal of child psychology and psychiatry, and allied disciplines'', ''58''(9), 1062–1064.</ref><ref>Norris, M., & Lecavalier, L. (2010). Screening accuracy of Level 2 autism spectrum disorder rating scales. A review of selected instruments. ''Autism : the international journal of research and practice'', ''14''(4), 263–284.</ref> |Mental Health |Social |Parent | | | |- |KSADS Diagnostic Interview for DSM-5 (Post-Traumatic Stress Disorder) - Parent<ref name=":6" /><ref name=":7" /><ref name=":2" /><ref name=":3" /> |Mental Health |Stress |Parent | | | |- |Life Events Scale - Parent<ref name=":10" /><ref name=":15" /> |Mental Health |Stress |Parent | | | |- |[[wikipedia:Perceived_Stress_Scale|Perceived Stress Scale]]<ref>Cohen, S., Kamarck, T., and Mermelstein, R. (1983). A global measure of perceived stress. Journal of Health and Social Behavior, 24, 386-396. </ref> |Mental Health |Stress |Parent | | | |- |Early Adolescent Temperament Questionnaire<ref>Latham MD, Dudgeon P, Yap MBH, Simmons JG, Byrne ML, Schwartz OS, Ivie E, Whittle S, Allen NB. Factor Structure of the Early Adolescent Temperament Questionnaire-Revised. Assessment. 2020 Oct;27(7):1547-1561. doi: 10.1177/1073191119831789. Epub 2019 Feb 21. PMID: 30788984.</ref> |Mental Health |Temperament/Personality |Parent | | | |- | | | | | | | |- |Brief Problem Monitor |Mental Health |Broad Psychopathology |Teacher | | | |- | | | | | | | |- |Behavioral Indicator of Resiliency to Distress Task (BIRD)<ref>Lejuez, C. W., Kahler, C. W., & Brown, R. A. (2003). A modified computer version of the Paced Auditory Serial Addition Task (PASAT) as a laboratory-based stressor. ''The Behavior Therapist, 26''(4), 290–293.</ref><ref>Feldner, M. T., Leen-Feldner, E. W., Zvolensky, M. J., & Lejuez, C. W. (2006). Examining the association between rumination, negative affectivity, and negative affect induced by a paced auditory serial addition task. ''Journal of behavior therapy and experimental psychiatry'', ''37''(3), 171–187.</ref> |Neurocognition |Tasks |Youth | | | |- |Cash Choice Task<ref>Wulfert, E., Block, J. A., Santa Ana, E., Rodriguez, M. L., & Colsman, M. (2002). Delay of gratification: impulsive choices and problem behaviors in early and late adolescence. ''Journal of personality'', ''70''(4), 533–552.</ref><ref>Anokhin, A. P., Golosheykin, S., Grant, J. D., & Heath, A. C. (2011). Heritability of delay discounting in adolescence: a longitudinal twin study. ''Behavior genetics'', ''41''(2), 175–183.</ref> |Neurocognition |Tasks |Youth | | | |- |Delay Discounting Task<ref>Johnson, M. W., & Bickel, W. K. (2008). An algorithm for identifying nonsystematic delay-discounting data. ''Experimental and clinical psychopharmacology'', ''16''(3), 264–274.</ref> |Neurocognition |Tasks |Youth | | | |- |Emotional Faces Stroop Task<ref>Başgöze, Z., Gönül, A. S., Baskak, B., & Gökçay, D. (2015). Valence-based Word-Face Stroop task reveals differential emotional interference in patients with major depression. ''Psychiatry research'', ''229''(3), 960–967.</ref><ref>Kane, M. J., & Engle, R. W. (2003). Working-memory capacity and the control of attention: the contributions of goal neglect, response competition, and task set to Stroop interference. ''Journal of experimental psychology. General'', ''132''(1), 47–70.</ref><ref>Stroop, J.R., 1935. Studies of interference in serial verbal reactions. J. Exp. Psychol. 18 (6), 643–662.</ref> |Neurocognition |Tasks |Youth | | | |- |Game of Dice Task<ref>Brand, M., Fujiwara, E., Borsutzky, S., Kalbe, E., Kessler, J., & Markowitsch, H. J. (2005). Decision-making deficits of Korsakoff patients in a new gambling task with explicit rules: associations with executive functions. ''Neuropsychology'', ''19''(3), 267–277.</ref><ref>Drechsler, R., Rizzo, P., & Steinhausen, H. C. (2008). Decision-making on an explicit risk-taking task in preadolescents with attention-deficit/hyperactivity disorder. ''Journal of neural transmission (Vienna, Austria : 1996)'', ''115''(2), 201–209.</ref><ref>Duperrouzel, J. C., Hawes, S. W., Lopez-Quintero, C., Pacheco-Colón, I., Coxe, S., Hayes, T., & Gonzalez, R. (2019). Adolescent cannabis use and its associations with decision-making and episodic memory: Preliminary results from a longitudinal study. ''Neuropsychology'', ''33''(5), 701–710.</ref><ref>Ross, J. M., Graziano, P., Pacheco-Colón, I., Coxe, S., & Gonzalez, R. (2016). Decision-Making Does not Moderate the Association between Cannabis Use and Body Mass Index among Adolescent Cannabis Users. ''Journal of the International Neuropsychological Society : JINS'', ''22''(9), 944–949.</ref> |Neurocognition |Tasks |Youth | | | |- |Little Man Task<ref>Acker, W. (1982). “A computerized approach to psychological screening—The Bexley-Maudsley Automated Psychological Screening and The Bexley-Maudsley Category Sorting Test.” ''International Journal of Man-Machine Studies'', ''17''(3): 361-369.</ref><ref>Nixon, S. J., Prather, R. A., & Lewis, B. (2014). Sex differences in alcohol-related neurobehavioral consequences. In Edith V. Sullivan and Adolf Pfefferbaum (Eds.), Alcohol and the nervous system (Handbook of clinical neurology, 3rd series (Vol. 125)). Oxford, United Kingdom, Elsevier, pp. 253-272.</ref> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Dimensional Change Card Sort<ref>{{Cite web|url=https://nihtoolbox.org/test/dimensional-change-card-sort-test/|title=Dimensional Change Card Sort Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Flanker Inhibitory Control & Attention<ref>{{Cite web|url=https://nihtoolbox.org/test/flanker-inhibitory-control-and-attention-test-age-12/|title=Flanker Inhibitory Control and Attention Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Oral Reading Recognition<ref>{{Cite web|url=https://nihtoolbox.org/test/a-dummy-iq-test/|title=Oral Reading Recognition Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Pattern Comparison Processing Speed<ref>{{Cite web|url=https://nihtoolbox.org/test/pattern-comparison-processing-speed/|title=Pattern Comparison Processing Speed Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Picture Sequence Memory<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-sequence-memory-test/|title=Picture Sequence Memory Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - Picture Vocabulary<ref>{{Cite web|url=https://nihtoolbox.org/test/picture-vocabulary-test/|title=Picture Vocabulary Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12">McDonald, Skye (Ed.) (2014). Special series on the Cognition Battery of the NIH Toolbox. ''Journal of International Neuropsychological Society'', 20 (6), 487-651.</ref> |Neurocognition |Tasks |Youth | | | |- |NIH Toolbox Tasks - List Sorting Working Memory<ref>{{Cite web|url=https://nihtoolbox.org/test/test/|title=List Sorting Working Memory Test|website=NIH Toolbox|language=en-US|access-date=2024-08-12}}</ref><ref name=":12" /> |Neurocognition |Tasks |Youth | | | |- |RAVLT Delayed Recall<ref name=":13" /><ref name=":14" /> |Neurocognition | |Youth | | | |- |Rey Auditory Verbal Leanring Task (RAVLT) <ref name=":13">Strauss, E., Sherman, E.M.S., & Spreen, O. (2006) A compendium of neuropsychological tests. Oxford University Press. New York, New York. Third Edition.</ref><ref name=":14">Lezak, M.D., Howieson, D.B., Bigler, E.D., & Tranel, D. (2012) Neuropsychological assessment. 5th Edition. Oxford University Press. New York, NY.</ref> |Neurocognition |Tasks |Youth | | | |- |Social Influence Task<ref>Knoll, L. J., Leung, J. T., Foulkes, L., & Blakemore, S. J. (2017). Age-related differences in social influence on risk perception depend on the direction of influence. ''Journal of adolescence'', ''60'', 53–63.</ref> |Neurocognition |Tasks |Youth | | | |- |Stanford Mental Arithmetic Response Time Evaluation (SMARTE)<ref>Starkey, G. S., & McCandliss, B. D. (2014). The emergence of “groupitizing” in children’s numerical cognition. ''Journal of experimental child psychology'', ''126'', 120–137.</ref> |Neurocognition |Tasks |Youth | | | |- |Wechsler Intelligence Scale for Children - Matrix Reasoning Task<ref>Wechsler, D. (2014). Wechsler Intelligence Scale for Children - Fifth Edition Manual. San Antonio,TX, Pearson.</ref><ref>Daniel, M.H., Wahlstrom, D. & Zhang, O. (2014) Equivalence of Q-interactive® and Paper Administrations of Cognitive Tasks: WISC®–V: Q-Interactive Technical Report.</ref> |Neurocognition |Tasks |Youth | | | |- | | | | | | | |- |Barkley Deficits in Executive Functioning Scale<ref>Barkley RA (2010). Differential diagnosis of adults with ADHD: the role of executive function and self-regulation. ''J Clin Psychiatry'', 71(7), e17. doi: 10.4088/JCP.9066tx1c</ref><ref>Barkley RA (2011). ''Barkley deficits in executive functioning scale (BDEFS for adults)''. New York: Guilford Press.</ref><ref>Barkley RA (2012). ''Barkley Deficits in Executive Functioning Scale--Children and Adolescents (BDEFS-CA)'': Guilford Press.</ref> |Neurocognition |Questionnaire |Parent | | | |- | | | | | | | |- |Alcohol Expectancies Questionnaire (AEQ-AB)<ref>Brown, S. A., Christiansen, B. A., & Goldman, M. S. (1987). The Alcohol Expectancy Questionnaire: an instrument for the assessment of adolescent and adult alcohol expectancies. ''Journal of studies on alcohol'', ''48''(5), 483–491.</ref><ref>Greenbaum, P. E., Brown, E. C., & Friedman, R. M. (1995). Alcohol expectancies among adolescents with conduct disorder: prediction and mediation of drinking. ''Addictive behaviors'', ''20''(3), 321–333.</ref><ref>Stein, L. A., Katz, B., Colby, S. M., Barnett, N. P., Golembeske, C., Lebeau-Craven, R., & Monti, P. M. (2007). Validity and Reliability of the Alcohol Expectancy Questionnaire-Adolescent, Brief. ''Journal of child & adolescent substance abuse'', ''16''(2), 115–127.</ref> |Substance Use |SU Attitude |Youth | | | |- |Alcohol Motives Questionnaire (PhenX)<ref>Cooper, M. L. (1994). Motivations for alcohol use among adolescents: Development and validation of a four-factor model. Psychological Assessment, 6, 117−128.</ref><ref>Grant, V. V., Stewart, S. H., O’Connor, R. M., Blackwell, E., & Conrod, P. J. (2007). 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''Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco'', ''7''(1), 81–90.</ref> |Substance Use |SU Attitude |Youth | | | |- |Electronic Nictotine Delivery Systems Expectancies Questionnaire<ref>Pokhrel, P., Lam, T.H., Pagano, I., Kawamoto, C.T., & Herzog, T.A. (2018). YPokhrel, P., Lam, T. H., Pagano, I., Kawamoto, C. T., & Herzog, T. A. (2018). Young adult e-cigarette use outcome expectancies: Validity of a revised scale and a short scale. ''Addictive behaviors'', ''78'', 193–199.</ref> |Substance Use |SU Attitude |Youth | | | |- |Electronic Nicotine Delivery System Motives Inventory<ref>Centers for Disease Control (CDC; Division of Nutrition). (2016). Anthropometry Procedures Manual.</ref><ref name=":24">Piper, M. E., Piasecki, T. M., Federman, E. B., Bolt, D. M., Smith, S. S., Fiore, M. C., & Baker, T. B. (2004). A multiple motives approach to tobacco dependence: the Wisconsin Inventory of Smoking Dependence Motives (WISDM-68). 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The life history calendar: a technique for collecting retrospective data. Sociological methodology, ''18'', 37–68.</ref> |Substance Use |SU Attitude |Youth | | | |- |PhenX Peer Tolerance of Use<ref name=":19" /><ref name=":20" /> |Substance Use |SU Attitude |Youth | | | |- |PhenX Perceived Harm of Substance Use<ref name=":19">Johnston, Lloyd D.; O’Malley, P. M.; Bachman, J. G.; Schulenberg, J. E.. (2009). Monitoring the Future. National Results on Adolescent Drug Use: Overview of Key Findings, 2009. NIH Publication Number 10-7583</ref><ref name=":20">Miech, R. A.; Johnston, L. D.; O’Malley, P. M.; Bachman, J. G.; Schulenberg, J. E.. (2015). Monitoring the Future National Survey Results on Drug Use, 1975-2014. Volume 1, Secondary School Students. Ann Arbor: Institute for Social Research: The University of Michigan.</ref> |Substance Use |SU Attitude |Youth | | | |- |Reasons for Electronic Nicotine Delivery Systems Use<ref name=":24" /><ref>Wills, T. A., Sandy, J. M., & Yaeger, A. M. 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International Journal of Environmental Research and Public Health, 14(3), 234-259.</ref> |Culture & Environment |Family |Youth | | | |- |PhenX [[wikipedia:Family_Environment_Scale|Family Environment Scale]] - Family Conflict - Youth<ref name=":32">Moos, R.H., Moos, B.S. (1994). Family Environment Scale Manual. Consulting Psychologists Press, Palo Alto, CA.</ref> |Culture & Environment |Family |Youth | | | |- |Acceptance Subscale from Children's Report of Parental Behavior Inventory (CRPBI) - Short<ref>Schaefer, E.S., 1965. A configurational analysis of children’s reports of parent behavior. J. Consult. Psychol. 29, 552–557.</ref><ref>Schludermann, E. H., & Schludermann, S. M. (1988). Children’s Report on Parent Behavior (CRPBI-108, CRPBI-30) for older children and adolescents. Winnipeg, MB, Canada: University of Manitoba.</ref><ref>Barber, B. K., Olsen, J. E., & Shagle, S. C. (1994). 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Psychological Assessment, 11(3),315‑325.</ref> |Culture & Environment |Family |Parent | | | |- |Parental Monitoring Survey<ref name=":30" /><ref name=":31" /> |Culture & Environment |Parenting |Parent | | | |- |School Attendance of Youth & Grades<ref name=":34" /> |Culture & Environment |School |Parent | | | |- |Prosocial Behavior Survey - Parent<ref name=":27" /> |Culture & Environment |Temperament/Personality |Parent | | | |- |HOME Short Form Cognitive Stimulation<ref>Bailey, C.T. & Boykin, A.W. (2001). The role of task variability and home contextual factors in the academic performance and task motivation of African American elementary school children. ''The Journal of Negro Education, 70''(1/2), 84-95. <nowiki>http://www.jstor.org/stable/2696285</nowiki></ref><ref>Boykin, A.W. & Cunningham, R.T. The effects of movement expressiveness in story content and learning context on the analogical reasoning performance of African American Children. ''Negro Education, 70''(1/2), 72-83. <nowiki>http://www.jstor.org/stable/2696284</nowiki></ref><ref>Zaslow, M. J., Weinfield, N. S., Gallagher, M., Hair, E. C., Ogawa, J. R., Egeland, B., ... & De Temple, J. M. (2006). Longitudinal prediction of child outcomes from differing measures of parenting in a low-income sample. ''Developmental Psychology'', ''42''(1), 27-37. <nowiki>https://doi.org/10.1037/0012-1649.42.1.27</nowiki></ref> |Culture & Environment | |Parent | | | |- | | | | | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Parent | | | |- |Fitbit - Pre/Post-Assessment Survey |Novel Technologies |Actigraphy |Parent | | | |- |EARS - Post-Assessment Survey |Novel Technologies |Screen Use |Parent | | | |- |EARS - Pre/Post-Assessment Survey (Pilot) |Novel Technologies |Screen Use |Parent | | | |- |Parent Screen Time Questionnaire |Novel Technologies |Screen Use |Parent | | | |} == Notes - Subscale Information == * For items without subscales, a general overview of variables that suggests ways that they may be sorted was given. * The ABCD Data Dictionary should be used to view administrative (non-item) variables, precedents for validation, and to further explore total and subscale score calculation. * Item variables were copied directly from the ABCD Data Dictionary. * Subscale total variables were listed as they were provided in the Data Dictionary. A subscale without a total score variable indicates that one was not included in the Data Dictionary. * A blank in any cell indicates that measure was mentioned in the ABCD Protocol by Wave PDFs, but could not be located in the Data Dictionary. * If a scale included items that were not listed as counting toward a specific subscale score, the corresponding variables were listed under "Other Items". Administrative variables (e.g., those pertaining to administration or missing data) were not included here. * Issues with variables in ABCD Data Dictionary ** The same variables were referred to by different names when listed independently and listed for subscale calulation in the PPS and the GISH. The spelling of these item variables should be verified. ** In, the data dictionary the notes for total severity score describes the variable as being the sum of all items assessing "How much did [this experience of psychosis] bother you" plus the "Did [this experience of psychosis] bother you?" subscale score. However, the variable label includes the sum of of all items assessing "How much did [this experience of psychosis] bother you" plus the number of questions asking "Did [this experience of psychosis] bother you?" that the child answered "no" to. It should be clarified whether the total of "no" responses or the total of "yes" responses should be used. ** Both modified and unmodified variables exist for all BIS/BAS scores, with the exception of BAS Fun Seeking. It is unclear what the difference between these variables is, and both the modified and unmodified versions were included in the table. == References == 0vwcdwxy06sdd6v26ad4ahadh8zamlq 0 313557 2691730 2691437 2024-12-13T02:08:59Z Dc.samizdat 2856930 removed duplicate footnote references 2691730 wikitext text/x-wiki {{Article info |journal=Wikijournal Preprints |last=Christie |first=David Brooks |abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-point 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} == Acknowledgements == This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations. == Future work == The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles. Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles. Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed. == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-point 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three coordinate systems.]] The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} == Acknowledgements == This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations. == Future work == The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles. Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles. Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed. == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-point 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three coordinate systems.{{Sfn|Egan|2019}}]] The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} == Acknowledgements == This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] and one by Greg Egan which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations. == Future work == The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles. Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles. Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed. == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-point 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.{{Sfn|Egan|2019}}]] The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} == Acknowledgements == This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] and one by Greg Egan which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations. == Future work == The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles. Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles. Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed. == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-point 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes. One octahedral cell of 24 is emphasized. Each octahedral cell has two antipodal vertices (one perpendicular axis) of each color: one axis from each of the three (w,x,y,z) coordinate systems.{{Sfn|Egan|2019}}]] The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} == Acknowledgements == This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] and one by Greg Egan which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations. == Future work == The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles. Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles. Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed. == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == === Wikipedia2Wikiversity === This page was based on the following [https://en.wikipedia.org/wiki/Holomorphic%20function wikipedia-source page]: * [https://en.wikipedia.org/wiki/Holomorphic%20function Holomorphic function] https://en.wikipedia.org/wiki/Holomorphic%20function * Datum: 11/4/2024 * [https://niebert.github./Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> rf04359qmfdr4gdsfy3ab24fhlb11g6 2691752 2691751 2024-12-13T07:49:00Z Bert Niehaus 2387134 /* Translation and Version Control */ 2691752 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with a [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == === Wikipedia2Wikiversity === This page was based on the following [https://en.wikipedia.org/wiki/Holomorphic%20function wikipedia-source page]: * [https://en.wikipedia.org/wiki/Holomorphic%20function Holomorphic function] https://en.wikipedia.org/wiki/Holomorphic%20function * Datum: 11/4/2024 * [https://niebert.github./Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> 7tr1ey8mo23pbg1eopr5mypeam1kpl8 2691753 2691752 2024-12-13T07:49:37Z Bert Niehaus 2387134 /* See also */ 2691753 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with a [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[/Kriterien/|Holomorphiekriterien]] * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == === Wikipedia2Wikiversity === This page was based on the following [https://en.wikipedia.org/wiki/Holomorphic%20function wikipedia-source page]: * [https://en.wikipedia.org/wiki/Holomorphic%20function Holomorphic function] https://en.wikipedia.org/wiki/Holomorphic%20function * Datum: 11/4/2024 * [https://niebert.github./Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> 0ykskbdl9exg1jt7dwms098yk3ktn92 2691754 2691753 2024-12-13T07:50:06Z Bert Niehaus 2387134 /* See also */ 2691754 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with a [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[/Criteria/|Holomorphic Criteria]] * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == === Wikipedia2Wikiversity === This page was based on the following [https://en.wikipedia.org/wiki/Holomorphic%20function wikipedia-source page]: * [https://en.wikipedia.org/wiki/Holomorphic%20function Holomorphic function] https://en.wikipedia.org/wiki/Holomorphic%20function * Datum: 11/4/2024 * [https://niebert.github./Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> j2k0yzn5gn91gqshw731k61q0bvz4ri 2691763 2691754 2024-12-13T08:22:48Z Bert Niehaus 2387134 /* Page Information */ 2691763 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with a [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[/Criteria/|Holomorphic Criteria]] * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function https://en.wikiversity.org/wiki/Holomorphic%20function] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> rkibijv1e2rkdapr3eee1locv7ppsrk 2691766 2691763 2024-12-13T08:42:37Z Bert Niehaus 2387134 /* Holomorphic Function */ 2691766 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with an [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[/Criteria/|Holomorphic Criteria]] * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function https://en.wikiversity.org/wiki/Holomorphic%20function] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> nqx4v8dqp3vgu5nx6xrosiyzp4l6ku4 2691768 2691766 2024-12-13T08:46:49Z Bert Niehaus 2387134 /* Holomorphic Function */ 2691768 wikitext text/x-wiki ==Holomorphic Function == '''Holomorphic''' (of [[w:en:Greek language|gr.]] ὅλος ''holos,'' 'whole' and μορφή ''morphe,'' 'form') is a property of certain [[w:en:complex valued function|complex valued functions]] which are analyzed in the [[Complex Analysis]] as a function <math display="inline"> f\colon U \rightarrow \mathbb{C} </math> with an [[w:en:open set|open set]] <math display="inline"> U \subseteq \mathbb{C} </math> is called holomorphic if <math>f</math> is [[w:en:Differentiable function|differentiable]] at each point of <math display="inline"> U </math>. In this topic also [[/Criteria/|equivalent criteria]] for a function being holomorphic are discussed. == Subtopics == * '''[[/Criteria/|Criteria]]''' - Equivalent criteria for a function being holomorphic ==Real and complex differentiation== Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a [[w:en:power series|power series]]. ===Definition: Complex differentiation=== It is <math display="inline"> U \subseteq \mathbb{C} </math> an open subset of the complex plane and <math display="inline"> z_0\in U </math> a point of this subset. A function <math display="inline"> f\colon U \to \mathbb{C} </math> is called ''complex differentiable'' in point <math display="inline"> z_0 </math>, if the [[w:en:limit (Funktion)|limit]] :<math display="block"> \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} </math> with <math display="inline"> h\in\mathbb{C}\setminus \{0\} </math>. If the limit exists, then the limit is denoted with <math display="inline"> f'(z_0) </math>. ===Definition: Holomorphic in one point=== Let <math display="inline"> U\subset \mathbb{C} </math> be an open set and <math display="inline"> f:U\to \mathbb{C} </math> a function. <math display="inline"> f </math> is called ''holomorphic'' in point <math display="inline"> z_0\in U </math>,'' if a [[w:en:neighbourhood (Mathematik)|neighbourhood]] of <math display="inline"> z_0 </math> exists, in which <math display="inline"> f </math> is complex differentiable. ===Definition: Full function=== If <math display="inline"> f:\mathbb{C}\to \mathbb{C} </math> is complexly differentiable to the whole <math display="inline"> \mathbb{C} </math>, then <math display="inline"> f </math> is called a ''[[w:en:ganze Funktion|entire function]].'' ==Explanatory notes== ===Link between complex and real differentiation=== <math display="inline"> \mathbb{C} </math> can be interpreted as a two-dimensional real vector space with the canonical base <math display="inline"> \{1, i\} </math> and so one can examine a function <math display="inline"> f\colon U \to \mathbb{C} </math> on an open set <math display="inline"> U \subseteq \mathbb{C} </math>. In [[w:en:Multivariable calculus|multivariable calculus]] it is known that <math display="inline"> f </math> [[w:en:total derivative|total]] [[w:en:Differentiable function|differentiable]] in <math display="inline"> z_0 </math> if there exists a <math display="inline"> \mathbb{R} </math>-linear mapping <math display="inline"> L </math>, so that :<math display="block"> f(z_0+h) = f(z_0) + L(h) + r(h) </math> where <math display="inline"> r </math> is a function with the property :<math display="block"> \lim_{h \to 0} \frac{r(h)}{|h|} = 0. </math> It can now be seen that the function <math display="inline"> f </math> is [[w:en:complex derivative|complex differentiable]] in <math display="inline"> z_0 </math>, if <math>f</math> is [[w:en:total derivative|total differentiable]] in <math display="inline"> z_0 </math> and <math display="inline"> L </math> is even <math display="inline"> \mathbb{C} </math>-linear. The latter is a [[w:en:Necessity and sufficiency|sufficient condition]]. It means that the [[w:en:Transformation matrix|transformation matrix]] <math display="inline"> L </math> with respect to the canonical base <math display="inline"> \{1, i\} </math> has the form :<math display="block"> L(z_1+i z_2) = C\left(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\cdot \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\right) </math> with <math display="inline"> C\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} := y_1 + iy_2 </math>. === Jacobi Matrix === : ''Main article: [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]]'' If a function <math display="inline"> f\left(x+iy\right)=u\left(x,y\right) + i\cdot v\left(x,y\right) </math> is decompose into functions of its real and imaginary parts with real-valued functions <math display="inline"> u, v </math>, the total derivative <math display="inline"> L </math> with [[w:en:transformation matrix|tranformation matrix]] has the [[w:en:Jacobian matrix and determinant|Jacobian matrix]] :<math display="block"> \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}. </math> ===Cauchy-Riemannian differential equations=== Consequently, the function <math display="inline"> f </math> is [[w:en:Total derivative|total differentiable]] precisely when it can be differentiated relatively and for <math display="inline"> u, v </math> the Cauchy-Riemann equations :<math display="block"> \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} </math> :<math display="block"> \displaystyle\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} </math> are fulfilled. ==Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:== * '''(H1)''' The function can be differentiated in a complex manner. * '''(H2)''' The function can be varied as often as desired. * '''(H3)''' Real and imaginary parts meet the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and can be continuously differentiated. * '''(H4)''' The function can be developed into a complex [[w:en:power series|power series]]. * '''(H5)''' The function is steady and the [[w:en:Kurvenintegral|path integral]] of the function disappears via any closed [[w:en:Homotopie|contractible]] path. * '''(H6)''' The functional values in the interior of a [[w:en:Kreisscheibe|circular disk]] can be determined from the functional values at the edge using the [[w:en:Cauchysche Integralformel|Cauchy Integral formula]]. * '''(H7)''' f can be differentiated and it applies <br /><math display="inline"> \quad\frac{\partial f}{\partial \bar z}=0, </math> <br />where <math display="inline"> \tfrac{\partial}{\partial \bar z} </math> the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\tfrac\partial{\partial\bar z} := \tfrac12\left(\tfrac\partial{\partial x}+i\tfrac\partial{\partial y}\right)</math> ==Examples== ===Entire Functions=== An [[w:en:Entire function|entire function]] is holomorphic on the whole <math display="inline"> \mathbb{C} </math>. Examples are: * each polynomial <math> \displaystyle z\mapsto\sum_{j=0}^na_jz^j </math> with coefficients <math display="inline"> a_j \in \mathbb{C} </math>, * the [[w:en:Exponential function|Exponential Function]] <math display="inline"> \exp </math>, * the [[w:en:Trigonometric Function|trigonometric functions]] <math display="inline"> \sin </math> and <math display="inline"> \cos </math>, * the [[w:en:Hyperbolic function|hyperbolic functions]] <math display="inline"> \sinh </math> and <math display="inline"> \cosh </math>. ===Holomorphic, non-gant functions=== * [[w:en:rational function|Rational functions]] are holomorphic apart from the zero points of polynomial in the denominator. Then [[w:en:rational function|rational function]] has [[w:en:isolated singularity|isolated singularities]] (e.g. [[w:en:Pole|poles]]. Rational functions are examples for [[w:en:meromorphic function|meromorphic functions]]. * The [[w:en:Logarithm|Logarithm function]] <math display="inline"> \log </math> can be developed at all points from <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math> into a power series and is thus holomorphic on the set <math display="inline"> \mathbb{C} \setminus {]{-\infty},0]} </math>. ===Functions - not holomorphic at any point === The following functions are not holomorphic in any <math display="inline"> z\in\mathbb{C} </math>. Examples are: * the absolute value function <math display="inline"> z\mapsto |z| </math>, * the projections on the real part <math display="inline"> z\mapsto\mathrm{Re}(z) </math> or on the imaginary part <math display="inline"> z\mapsto\mathrm{Im}(z) </math>, * complex conjugation <math display="inline"> z\mapsto\overline{z} </math>. The function <math display="inline"> z \mapsto |z|^2 </math> is complex differentiable only at the point <math display="inline"> z_o = 0 </math>, but the function is ''not''' holomorphic in <math>z_o</math>, since it is not complex differentiable in a neighborhood of <math display="inline"> 0 </math>. == Properties == Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<ref> {{cite book | last = Henrici | first = Peter | author-link = w:en:Peter Henrici (mathematician) | year = 1993 | orig-year = 1986 | title = Applied and Computational Complex Analysis | volume = 3 | place = New York - Chichester - Brisbane - Toronto - Singapore | publisher = [[w:en:John Wiley & Sons|John Wiley & Sons]] | series = Wiley Classics Library | edition = Reprint | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1 | url = https://books.google.com/books?id=vKZPsjaXuF4C |via=Google }} </ref> That is, if functions <math> f</math> and <math> g</math> are holomorphic in a domain <math> U</math> , then so are <math> f+g</math> , <math> f-g</math> , <math> fg</math> , and <math> f \circ g</math> . Furthermore, <math> f/g </math> is holomorphic if <math> g</math> has no zeros in <math> U</math> ; otherwise it is [[w:en:meromorphic|meromorphic]]. If one identifies <math> \C</math> with the real [[w:en:plane (geometry)|plane]] <math> \textstyle \R^2</math> , then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[w:en:Cauchy–Riemann equations|Cauchy–Riemann equations]], a set of two [[w:en:partial differential equation|partial differential equation]]s.<ref name=Mark> Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.</ref> === Functions for real and imaginary parts === Every holomorphic function can be separated into its real and imaginary parts <math> 1=f(x + iy) = u(x, y) + i\,v(x,y)</math> , and each of these is a [[w:en:harmonic function|harmonic function]] on <math> \textstyle \R^2</math> (each satisfies [[w:en:Laplace's equation|Laplace's equation]] <math> 1=\textstyle \nabla^2 u = \nabla^2 v = 0</math> ), with <math> v</math> the [[w:en:harmonic conjugate|harmonic conjugate]] of <math> u</math> .<ref> {{cite book |first=L.C. |last=Evans |author-link=w:en:Lawrence C. Evans |year=1998 |title=Partial Differential Equations |publisher=American Mathematical Society }} </ref> Conversely, every harmonic function <math> u(x, y)</math> on a [[w:en:Simply connected space|simply connected]] domain <math> \textstyle \Omega \subset \R^2</math> is the real part of a holomorphic function: If <math> v</math> is the harmonic conjugate of <math> u</math> , unique up to a constant, then <math> 1=f(x + iy) = u(x, y) + i\,v(x, y)</math> is holomorphic. === Cauchy's integral theorem === [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] implies that the [[w:en:contour integral|contour integral]] of every holomorphic function along a [[w:en:loop (topology)|loop]] vanishes:<ref name=Lang> {{cite book |first = Serge |last = Lang | author-link = w:en:Serge Lang | year = 2003 | title = Complex Analysis | series = Springer Verlag GTM | publisher = [[w:en:Springer Verlag|Springer Verlag]] }} </ref> :<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math> Here <math> \gamma</math> is a [[w:en:rectifiable path|rectifiable path]] in a simply connected [[w:en:domain (mathematical analysis)|complex domain]] <math> U \subset \C</math> whose start point is equal to its end point, and <math> f \colon U \to \C</math> is a holomorphic function. === Cauchy's integral formula === [[w:en:Cauchy's integral formula|Cauchy's integral formula]] states that every function holomorphic inside a [[w:en:disk (mathematics)|disk]] is completely determined by its values on the disk's boundary.<ref name=Lang/> Furthermore: Suppose <math> U \subset \C</math> is a complex domain, <math> f\colon U \to \C</math> is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \le r \} </math> is [[w:en:neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in <math> U</math> . Let <math> \gamma</math> be the circle forming the [[w:en:boundary (topology)|boundary]] of <math> D</math> . Then for every <math> a</math> in the [[w:en:interior (topology)|interior]] of <math> D</math> : :<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math> where the contour integral is taken [[w:en:curve orientation|counter-clockwise]]. === Cauchy's differentiation formula === The derivative <math> {f'}(a)</math> can be written as a contour integral<ref name=Lang /> using [[w:en:Cauchy's differentiation formula|Cauchy's differentiation formula]]: :<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math> for any simple loop positively winding once around <math> a</math> , and :<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math> for infinitesimal positive loops <math> \gamma</math> around <math> a</math> . === Conformal map === In regions where the first derivative is not zero, holomorphic functions are [[w:en:conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref> {{cite book | last =Rudin | first =Walter | author-link = w:en:Walter Rudin | year=1987 | title=Real and Complex Analysis | publisher=McGraw–Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 }} </ref> === Analytic - Taylor series === Every [[w:en:holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function <math> f</math> has derivatives of every order at each point <math> a</math> in its domain, and it coincides with its own [[w:en:Taylor series|Taylor series]] at <math> a</math> in a neighbourhood of <math> a</math> . In fact, <math> f</math> coincides with its Taylor series at <math> a</math> in any disk centred at that point and lying within the domain of the function. === Functions as complex vector space === From an algebraic point of view, the set of holomorphic functions on an open set is a [[w:en:commutative ring|commutative ring]] and a [[w:en:complex vector space|complex vector space]]. Additionally, the set of holomorphic functions in an open set <math> U</math> is an [[w:en:integral domain|integral domain]] if and only if the open set <math> U</math> is connected. <ref name="Gunning"> Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601</ref> In fact, it is a [[w:en:locally convex topological vector space|locally convex topological vector space]], with the [[w:en:norm (mathematics)|seminorms]] being the [[w:en:suprema|suprema]] on [[w:en:compact subset|compact subset]]s. === Geometric perspective - infinitely differentiable === From a geometric perspective, a function <math> f</math> is holomorphic at <math> z_0</math> if and only if its [[w:en:exterior derivative|exterior derivative]] <math> \mathrm{d}f</math> in a neighbourhood <math> U</math> of <math> z_0</math> is equal to <math> f'(z)\,\mathrm{d}z</math> for some continuous function <math> f'</math> . It follows from :<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math> that <math> \mathrm{d}f' </math> is also proportional to <math> \mathrm{d}z</math> , implying that the derivative <math> \mathrm{d}f'</math> is itself holomorphic and thus that <math> f</math> is infinitely differentiable. Similarly, <math> 1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0</math> implies that any function <math> f</math> that is holomorphic on the simply connected region <math> U</math> is also integrable on <math> U</math> . === Choice of Path - Independency === For a path <math> \gamma</math> from <math> z_0</math> to <math> z</math> lying entirely in <math> U</math> , define <math> 1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z </math> ; in light of the [[w:en:Jordan curve theorem|Jordan curve theorem]] and the [[w:en:Stokes' theorem|generalized Stokes' theorem]], <math> F_\gamma(z)</math> is independent of the particular choice of path <math> \gamma</math> , and thus <math> F(z)</math> is a well-defined function on <math> U</math> having <math> 1= \mathrm{d}F = f\,\mathrm{d}z</math> or <math> 1= f = \frac{\mathrm{d}F}{\mathrm{d}z} </math> . ==Biholomorphic functions== A function which is holomorphous [[w:en:Biholomorphic function|bijective]] and whose reverse function is holomorph is called ''biholomorph.'' In the case of a complex change, the equivalent is that the image is bijective and [[w:en:Conformal map|conformal]]. From the [[w:en:Implicit function theorem|Implicit Function Theorem]] it implies for holomorphic functions of a single variable that a [[w:en:Bijection|bijective]], [[w:en:holomorphic function|holomorphic function]] always has a holomorphic inverse function. ==Holomorphy of several variable== ===In the n-dimensional complex space=== Let<math display="inline"> D \subseteq \mathbb{C}^n </math> a complex open subset. An illustration <math display="inline"> f \colon D \to \mathbb{C}^m </math> is called holomorph if <math display="inline"> f = (f_1, \dotsc, f_m) </math> is holomorphous in each sub-function and each variable. With the [[w:en:Wirtinger-Kalkül|Wittgenstein-calculus]] <math display="inline"> \textstyle \frac{\partial}{\partial z^j} </math> and <math display="inline"> \textstyle \frac{\partial}{\partial \overline{z}^j} </math> a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply <math display="inline"> f \colon D \to \mathbb{C} </math> and the [[w:en:Identitätssatz für holomorphic functionen|Identity law]] is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by [[w:en:Induktion (Mathematik)|Induction]] to <math display="inline"> n </math> dimensions. [[w:en:Salomon Bochner|Salomon Bochner]] even proved in 1944 a generalization of the <math display="inline"> n </math>-dimensional Cauchy integral formula. This bears the name [[w:en:Bochner-Martinelli-Formel|Bochner-Martinelli-Formel]]. ===In complex geometry=== Holomorphic images are also considered in the [[w:en:Komplexe Geometrie|Complex Geometry]]. Thus, holomorphic images can be defined between [[w:en:Riemannsche Fläche|Riemann surface]] and between [[w:en:Komplexe Mannigfaltigkeit|Complex Manifolds]] analogously to differentiable functions between [[w:en:Glatte Mannigfaltigkeit|smooth maifolds]]. In addition, there is an important counterpart to the [[w:en:Differentialform|smooth Differential forms]] for integration theory, called [[w:en:holomorphe Differentialform|holomorphic Differential form]]. ==Literature== * Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer. * Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media. == References == <references/> ==See also== * [[/Criteria/|Holomorphic Criteria]] * [[w:en:Holomorph (mathematics)|Holomorph Group]] * [[Complex Analysis|Complex Analysis]] * [[w:en:Holomorphic function|holomorphic function]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function https://en.wikiversity.org/wiki/Holomorphic%20function] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphe%20Funktion Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[w:de:Holomorphie|Holomorphie]] URL: https://de.wikiversity.org/wiki/Holomorphie * Date: 11/4/2024 <span type="translate" src="Holomorphie" srclang="de" date="11/4/2024" time="14:38" status="inprogress"></span> <noinclude> [[de:Holomorphe Funktion]] </noinclude> 00q1tot7j2yaon7kx9gn3c0qftc9srn 2 316552 2691731 2691574 2024-12-13T05:49:23Z Jaredscribe 2906761 /* DHHS */ MAHA man, Economist 9 Nov 2691731 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission|founder=Donald Trump|extinction=4 July 2026 (planned)|mission=(In the words of president-elect Donald Trump: * dismantle government bureaucracy * slash excess regulations * cut wasteful expenditures * restructure federal agencies, * address "massive waste and fraud" in government spending}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]],<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref> with support from many [[w:Political_appointments_of_the_second_Trump_administration|Political and cabinet appointees of the second Trump administration]] and from a Congressional caucus Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == President-elect Trump has promised to "put an end to endless wars", to make [[w:NATO#NATO_defence_expenditure|NATO members pay their fair share]], end the [[w:Russian_invasion_of_Ukraine|current Russian invasion of Ukraine]], to renew the maximum-pressure policy toward Iran, and to free the hostages held in Gaza and/or ensure Israeli victory in the [[w:Israel–Hamas_war|current multi-front war launched by Iran and its proxies]]. NATO Secretary General [[w:Mark_Rutte|Mark Rutte]] publicly thanked Trump for stimulating Europe to increase national defense spending above 2%, saying "this is his doing, his success, and we need to do more, we notice."<ref>{{Cite news|url=https://www.wsj.com/video/wsj-opinion-twilight-of-the-trans-atlantic-relationship/FA4C937B-57AF-4E1D-BAC4-7293607577D1?page=1|title=WSJ Opinion: Twilight of the Trans-Atlantic Relationship|last=WSJ Opinion|date=26 November 2024|work=The Wall Street Journal}}</ref> Nominee for [[w:National_Security_Advisor|National Security Advisor]] [[w:Mike_Waltz|Mike Waltz]] To oversee the [[w:U.S._Intelligence_Community|U.S. Intelligence Community]] and NIP, and the 18 IC agencies, including the CIA, DIA, NSC, the nominee for [[w:Director_of_National_Intelligence|Director of National Intelligence]] is [[w:Tulsi_Gabbard|Tulsi Gabbard]], who is an isolationist of the [[w:Bernie_Sanders#foreign_policy|Bernie Sanders]] camp, with a long record of dogmatically opposing [[w:Foreign_policy_of_the_Trump_administration|President Trump's first term foreign policy]].<ref>{{Cite news|url=https://www.wsj.com/opinion/tulsi-gabbard-director-of-national-intelligence-donald-trump-foreign-policy-syria-israel-iran-b37aa3de|title=How Tulsi Gabbard Sees the World|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}}</ref> ''"The first act of a statesman is to recognize the type of war he is in"'', according to [[w:Carl_von_Clausewitz|Clausewitz]], given that human determination outweighs material advantages. Therefore he is advised by [[w:West_Point|West Point]] strategist [[w:John_Spencer|John Spencer]] writing in the WSJ to avoid four common foreign-policy fallacies: * the "abacus fallacy" that wars are won by superior resources, counterexample Vietnam * the "vampire fallacy" that wars are won by superior technology, counterexample Russia's failure in Ukraine, (c.f. Lt. Gen [[w:H.R._McMaster|H.R. McMaster]], 2014) * the "Zero Dark Thirty" fallacy that elevates precision strikes and special ops to the level of grand strategy or above (ibid) * and the "Peace table fallacy", which believes that all wars end in negotiation.<ref>{{Cite news|url=https://www.wsj.com/opinion/stopping-endless-wars-is-easier-said-than-done-trump-second-term-2cab9c7a?page=1|title=Stopping ‘Endless Wars’ Is Easier Said Than Done|last=Spencer|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> === Department of State === {{Main article|w:Second presidency of Donald Trump#Prospective foreign policy|w:State Department}} [[w:Marco_Rubio|Marco Rubio]] has been nominated as [[w:U.S._Secretary_of_State|U.S. Secretary of State]], overseeing $53bn and 77,880 employees ==== [[w:USAID|USAID]] ==== ==== National Endowment for Democracy ==== The [[w:National_Endowment_for_Democracy|National Endowment for Democracy]] is grant-making foundation organized as a private non-profit corporation overseen by congress, a project of Ronald Reagan announced in a [[1982 speech to British Parliament]], in which he stated that "freedom is not the sole prerogative of a lucky few, but the inalienable and universal right of all human beings", invoking the Israelites exodus and the Greeks' stand at Thermopylae. The NED is reportedly near the top of the DOGE's hit list.<ref>{{Cite news|url=https://www.wsj.com/opinion/save-a-reagan-initiative-from-the-doge-national-endowment-for-democracy-funding-2b6cc072?page=1|title=Save a Reagan Initiative From Musk and Ramaswamy|last=Galston|first=William|date=11 December 2024|work=The Wall Street Journal}}</ref> It is had a pro-freedom and [[w:Anti-communist|anti-communist]] mission to help pro-democracy leaders and groups in Asia, Africa, and Latin America, and assisted the transition of Eastern and Central European nations. The arguments being made by those in favor of defunding are that it "is a relic of the Cold War that has outlived its usefulness and no longer serves any pressing purpose in terms of advancing national interests", according to [[w:James_Piereson|James Piereson]]{{Cn}} Congress has raised its funding significanty in recent years, in a vote of confidence. === U.S. Department of Defense === U.S. DoD employees ____ civilian personel, ___ civilian contractors, and oversees a budget of _____. Nominated for [[w:Secretary_of_Defense|Secretary of Defense]] is [[w:Pete_Hegseth|Pete Hegseth]], who has been doubted by many Republican Senators{{Cn}} and supported by Trump's base.{{Cn}} The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === [[w:DARPA|DARPA]] === === US Air Force === Air Force is advancing a program called [[w:Collaborative_Combat_Aircraft|Collaborative Combat Aircraft]] to build roughly 1,000 UAVs, with [[w:Anduril|Anduril]] and [[w:General_Atomics|General Atomics]] currently building prototypes, ahead of an Air Force decision on which company or companies will be contracted to build it. The cost quickly exceeded the $2.3 billion approved for last fiscal year’s budget, according to the [[w:Congressional_Research_Service|Congressional Research Service]], prompting calls for more oversight. “''If you want to make real improvements to the defense and security of the United States of America, we would be investing more in drones, we’d be investing more in [[w:Hypersonic_weapon|hypersonic missiles]]'',” said Mr. Ramaswamy. The program for Lockheed-Martin's [[w:F-35|F-35]] stealth jet fighters, now in production, is expected to exceed $2 trillion over several decades. The Air Force on 5 December announced it would delay a decision on which company would build the next-generation crewed fighter, called [[w:Next Generation Air Dominance|Next Generation Air Dominance (NGAD)]], which was planned to replace the [[w:F-22|F-22]] and operate alongside the F-35. Mr. Musk has written that "manned fighter jets are obsolete in the age of drones.” In another post, he claimed “a reusable drone” can do everything a jet fighter can do “without all the overhead of a pilot.” Brigadier General [[w:Doug_Wickert|Doug Wickert]] said in response, “There may be some day when we can completely rely on roboticized warfare but we are a century away.... How long have we thought full self-driving was going to be on the Tesla?” <ref>{{Cite news|url=https://www.wsj.com/politics/national-security/air-force-jets-vs-drones-trump-administration-8b1620a5?page=1|title=Trump Administration Set to Decide Future of Jet Fighters|last=Seligman|first=Lara|date=6 December 2024|work=The Wall Street Journal}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Space Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. === Rail and Tunnel Authority === === Ports Authority === === Interstate Highway Authority === === [[w:Federal_Aviation_Administration|Federal Aviation Administration]] === Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} === National Air and Space Administration === The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] (NASA) had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization." In an interview Isaacman said that NASA will evolve as private space companies set their own priorities and develop technology. NASA could have a certification role for astronauts and vehicles, similar to how the Federal Aviation Administration oversees the commercial airline industry. “The FAA doesn’t build the airplanes. They don’t staff the pilots that fly you from point A to B,” he said. “That is the world that NASA is in, essentially.” He also suggested openness to new and lower cost ways of getting to the Moon and to Mars.<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} === National Oceanic and Atmospheric Administration === == Department of Education and Propaganda == [[w:United_States_Department_of_Education|Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> Nominated as [[w:US_Secretary_of_Education|Secretary of Education]] is [[w:Linda_McMahon|Linda McMahon]]. The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> During his campaign, Trump had pledged to get the "transgender insanity the hell out of schools.” Relying on the district court's decision in ''[[w:Tatel_v._Mount_Lebanon_School_District|Tatel v. Mount Lebanon School District]] , the'' attorney general and education secretary could issue a letter explaining how enforcing gender ideology violates constitutional [[w:Free_exercise_clause|First amendment right to free exercise of religion]] and the [[w:Equal_Protection_Clause|14th Amendment’s Equal Protection Clause]].<ref>{{Cite news|url=https://www.wsj.com/opinion/how-trump-can-target-transgenderism-in-schools-law-policy-education-369537a7?page=1|title=How Trump Can Target Transgenderism in Schools|last=Eden|first=Max|date=9 December 2024|work=The Wall Street Journal}}</ref> === CPB, PBS, NPR === Regarding the [[w:Corporation_for_Public_Broadcasting|Corporation for Public Broadcasting]], [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> {{As of|2024|alt=For fiscal year 2024}}, its appropriation was US$525 million, including $10 million in interest earned. The distribution of these funds was as follows:<ref>{{cite web|url=https://cpb.org/aboutcpb/financials/budget/|title=CPB Operating Budget|last=|date=2024|website=www.cpb.org|archive-url=|archive-date=|access-date=November 27, 2024|url-status=}}</ref> * $262.83M for direct grants to local public television stations; * $95.11M for television programming grants; * $81.77M for direct grants to local public radio stations; * $28.12M for the Radio National Program Production and Acquisition * $9.43M for the Radio Program Fund * $31.50 for system support * $26.25 for administration Public broadcasting stations are funded by a combination of private donations from listeners and viewers, foundations and corporations. Funding for public television comes in roughly equal parts from government (at all levels) and the private sector.<ref>{{cite web|url=http://www.cpb.org/annualreports/2013/|title=CPB 2013 Annual Report|website=www.cpb.org|archive-url=https://web.archive.org/web/20160212170045/http://cpb.org/annualreports/2013/|archive-date=February 12, 2016|access-date=May 4, 2018|url-status=dead}}</ref> == Department of Justice (DOJ) == The US. [[w:Department_of_Justice|Department of Justice]] has a 2023 budget of _____ and ___ employees. Nominated as [[w:Attorney_General|Attorney General]] is Florida AG [[w:Pam_Bondi|Pam Bondi]]{{Cn}}, after Matt Gaetz withdrew his candidacy after pressure.{{Cn}} === Federal Bureau of Investigation (FBI) === With 35,000 employees the [[w:FBI|FBI]] made a 2021 budget request for $9.8 billion. Nominated as director is [[w:Kash_Patel|Kash Patel]], who promised to "shut down the FBI [[w:Hoover_building|Hoover building]] on day one, and open it the next day as a museum of the deep state. He said "''I would take the 7,000 employees that work in that building and send them out across the America to chase criminals"'', saying ''"Go be cops."'' He promised to retaliate against journalists and government employees who "helped Joe Biden rig the election" in 2020.<ref>{{Cite web|url=https://www.wsj.com/video/series/wsj-explains/who-is-kash-patel-donald-trumps-pick-to-lead-the-fbi/F4D38D41-013A-4B05-A170-D7394AA91C2B|title=Who Is Kash Patel, Donald Trump’s Pick to Lead the FBI?|date=6 December 2024|website=WSJ.com Video}}</ref> ===== Reception and Analysis ===== His nomination "sent shock waves" through the DOJ, and his nomination has been opposed by many Republican lawmakers{{Cn}}, including former CIA director [[w:Gina_Haspell|Gina Haspell]] and AG [[w:Willliam_Barr|Willliam Barr]], who had threatened to resign if Mr. Patel were to be forced on them as a deputy, during Mr. Trump's first term.<ref name=":6">{{Cite news|url=https://www.wsj.com/opinion/kash-patel-doesnt-belong-at-the-fbi-cabinet-nominee-5ef655eb?page=1|title=Kash Patel Doesn’t Belong at the FBI: At the NSC, he was less interested in his assigned duties than in proving his loyalty to Donald Trump.|last=Bolton|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> As an author he wrote a polemical children's book lionizing "King Donald" with himself in the role of "Wizard", despite the Constitution's republican values and its prohibition on granting titles of nobility.<ref>Original synthethic statement, with constitutional context provided by <nowiki>[[User:Jaredscribe]]</nowiki></ref> He has been accused of exaggerating his roles and accomplishments, and deliberate vowing to violate the [[w:Article_Two_of_the_United_States_Constitution#Clause_5:_Caring_for_the_faithful_execution_of_the_law|"Take care" clause of Article II.3]], "''that the Laws be faithfully executed"'' by placing personal loyalties, vendettas, and hunches above his oath to the Constitution.<ref name=":6" /> He has also been accused of lying about national intelligence by [[w:Mark_Esper|Mark Esper]] in his memoir, recently again by Pence aide [[w:Olivia_Troye|Olivia Troye]], although former SoS [[w:Mike_Pompeo|Mike Pompeo]] has not yet clarified the incident in question.  He was called upon to do so in a 11 December WSJ piece by former National Security Advisor [[w:John_Bolton|John Bolton]], who also wrote, "''If illegitimate partisan prosecutions were launched [by the Biden administration], then those responsible should be held accountable in a reasoned, professional manner, not in a counter-witch hunt.  The worst response is for Mr. Trump to engage in the prosecutorial [mis]conduct he condemns [which further] politicizes and degrades the American people's faith in evenhanded law enforcement.''"<ref name=":6" /> He has received support from _____ who wrote that _______.{{Cn}} == U.S. Department of Health and Human Services (DHHS) == Nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United States Secretary of Health and Human Services]] is [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]], deputy Secretary [[w:Jim_O'Neill_(investor)|Jim O'Neill]]. Mr. Kennedy warned on 25 October that the FDA's "war on public health is about to end", accusing it of suppressing psychedelics, stem cells, raw milk, hydroxycloroquine, sunshine, and "anything else that advances human health and can't be patented by Pharma." He said that on day one he would "advise all US water systems to remove fluoride from public water".<ref>{{Cite news|title=MAHA man|date=9 November 2024|work=The Economist|department=News editors}}</ref> Nominated for [[w:US_Surgeon_General|US Surgeon General]] is [[w:Janette_Nesheiwat|Janette Nesheiwat]]. [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Nominee !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Marty_Makary|Marty Makary]] |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- | |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- | |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Dave_Weldon|Dave Weldon]] |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:Jay_Bhattacharya|Jay Bhattacharya]] |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- | |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- | |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Mehmet_Oz|Mehmet Oz]] |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- | |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- | |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === The [[w:NIH|NIH]] distributes grants of ~$50bn per year. Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]], who has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic", due to a phenomenon has been called [[Eroom’s law]], which explains that career incentives encourage “me-too research,” given that citations by other scientists “have become the dominant way to evaluate scientific contributions and scientists.” That has shifted research “toward incremental science and away from exploratory projects that are more likely to fail, but which are the fuel for future breakthroughs.”<ref name=":3">{{Cite news|url=https://www.wsj.com/opinion/jay-bhattacharya-and-the-vindication-of-the-fringe-scientists-pandemic-lockdowns-38b6aec6|title=Jay Bhattacharya and the Vindication of the ‘Fringe’ Scientists|last=Finley|first=Allysia|date=1 December 2024|work=Wall Street Journal}}</ref> Dr. Bhattacharya's February 2020 paper explaining Eroom's law, as possible explanation for slowing of pharmaceutical advances.{{Cn}} * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]], encouraging academic freedom among NIH scientists and term limits for NIH leaders. “Those kinds of reforms, I think every scientist would agree, every American would agree, it’s how you turn the NIH from something that is sort of how to control society, into something that is aimed at the discovery of truth to improve the health of Americans,” he said.<ref name=":4">{{Cite news|url=https://www.wsj.com/health/healthcare/covid-lockdown-critic-jay-bhattacharya-chosen-to-lead-nih-2958e5e2?page=1|title=Covid-Lockdown Critic Jay Bhattacharya Chosen to Lead NIH|last=Whyte|first=Liz Essley|date=26 November 2024|work=The Wall Street Journal}}</ref> * Refocusing on research on [[w:Chronic_diseases|chronic diseases]], which is underfunded, and away from [[w:Infectious_diseases|infectious diseases]], which is overfunded. * Ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague [[w:Eran_Bendavid|Eran Bendavid]], arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}&lt;nowiki&gt;</ref> Dr. Bhattacharya, [[w:Martin_Kulldorff|Martin Kulldorff]], then at Harvard, and Oxford’s [[w:Sunetra_Gupta|Sunetra Gupta]] formally expounded this idea in the [[w:Great_Barrington_Declaration|Great Barrington Declaration]] in October 2020, urging the government to focus on protecting the vulnerable while letting others go about their lives, which previous NIH director [[w:Francis_Collins|Francis Collins]] derided as "fringe science its into the political views of certain parts of our confused political establishment," and previous [[w:NIAID|NIAID]] director [[w:Chief_Medical_Advisor_to_the_President|chief medical advisor to the President]] [[w:Anthony_Fauci|Anthony Fauci]] "a quick and devastating public takedown of its premises." Some suggest the same career incentives that lead to scientific group-think in the pharmaceutical industry, also explain conformist behavior during COVID-19, due to the threat against young scientists of losing NIH funding, jobs, and career opportunities, if they were to exercise in independent judgement.<ref name=":3" /> “Dr. Jay Bhattacharya is the ideal leader to restore NIH as the international template for gold-standard science and evidence-based medicine,” DHHS Secretary nominee Kennedy wrote. "We will reform American scientific institutions so that they are worthy of trust again and will deploy the fruits of excellent science to make America healthy again!” said Dr. Bhattacharya. “Dr. Bhattacharya is a strong choice to lead the NIH,” said Dr. [[w:Ned_Sharpless|Ned Sharpless]], a former [[w:National_Cancer_Institute|National Cancer Institute]] director. “The support of moderate Senate Republicans will be critical to NIH funding, and Dr. Bhattacharya’s Covid work will give him credibility with this constituency.”<ref name=":4" /> === Food and Drug Administration === The FDA in 2022 had 18,000 employees<ref name="fy2022">{{cite web|url=https://www.fda.gov/media/149613/download|title=FY 2022 FDA Budget Request|publisher=FDA|archive-url=https://web.archive.org/web/20230602090805/https://www.fda.gov/media/149613/download|archive-date=June 2, 2023|access-date=January 14, 2022|url-status=live}}</ref> and a budget of $6.5{{nbsp}}billion (2022)<ref name="fy2022" /> Nominated as director is [[w:Marty_Makary|Marty Makary]]. == Department of Agriculture and Food == [[w:Department_of_Agriculture|Department of Agriculture]] (USDA) had 2023 budget of ___ and ____ employees. Nominated for [[w:Secretary_of_Agriculture|Secretary of Agriculture]] is [[w:Brooke_Rollins|Brooke Rollins]], who had earlier served on the [[w:Office_of_American_Innovation|Office of American Innovation]] under [[w:Jared_Kushner|Jared Kushner]], and served as director of [[w:Domestic_Policy_Council|Domestic Policy Council]]. She has not endorsed the "[[w:Make_America_Healthy_Again|Make America Healthy Again]]" agenda of RFK Jr. (and his colleagues Jay Bhattacharcya and others) who promised to "reverse 80 years of farm policy" and complains of the $30 billion/year farms subsidies. Kennedy wants to remove soda from food aid, and ultra-processed food from both [[w:Food-stamps|food-stamp]] benefits and [[w:School_meals|school meals]], both of which are overseen by the USDA; an effort that in the past has been opposed by the food industry, lawmakers, and some anti-hunger advocacy groups.<ref name=":5" /> RFK Jr.'s team had recommended [[w:Sid_Miller|Sid Miller]] for the role, and a group of farmers he had asked to vet candidates had proposed [[w:John_Kempf|John Kempf]].<ref name=":5">{{Cite news|url=https://www.wsj.com/politics/policy/trump-agriculture-pick-brooke-rollins-rfk-jr-1a85beda?page=1|title=RFK Jr. Team Skeptical About USDA Pick|last=Andrews|first=Natalie|date=11 December 2024|work=The Wall Street Journal|others=et al}}</ref> He has also called for re-examining the the [[standards regulating the use of pesticides]], especially [[w:Glyphosate|glyphosate]], the world's most widely used [[w:Herbicide|herbicide]] and the active ingredient in [[w:Roundup|Roundup]], used as a weedkiller in major [[w:U.S._commodity_crops|U.S. commodity crops]].[[w:Herbicide|herbicide]]<ref name=":5" /> === Food Stamps === === School Meals === === Farm Subsidies === == Department of Treasury and Reserve == [[w:Departments_of_Treasury|Departments of Treasury]] has 2023 budget of ____ and ___ employees. Nominated for [[w:Secretary_of_the_Treasury|Secretary of the Treasury]] is [[w:Scott_Bessent|Scott Bessent]]. === Consumer Financial Protection Bureau === The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> === Securities and Exchange Commission === [[w:Securities_and_Exchange_Commission|Securities and Exchange Commission]] === Internal Revenue Service === [[w:Internal_Revenue_Service|Internal Revenue Service]] === Federal Reserve === Mr. Musk has suggested starting a "[[w:Sovereign_Wealth_Fund|Sovereign Wealth Fund]]" like Texas and other U.S. states, instead of hosting a [[w:National_debt|national debt]]. Ron Paul and others have called for abolishing America's [[w:Central_Bank|Central Bank]], the [[w:Federal_Reserve|Federal Reserve System]], which Mr. Musk appeared to endorse. === American Sovereign Wealth Fund === == Department of Industry, Labor, and Commerce == [[w:Department_of_Commerce|Department of Commerce]] has 2023 budget of _____ and _____ employees. Nominated as [[w:Secretary_of_Commerce|Secretary of Commerce]] is [[w:Howard_Lutnick|Howard Lutnick]] [[w:Department_of_Labor|Department of Labor]] has 2023 budget of _____ and ____ employees. Nominated as [[w:Secretary_of_Labor|Secretary of Labor]] is [[w:Lori_Chavez-Remer|Lori Chavez-Remer]] == Departments of Energy and Interior == Nominee for [[w:US_Secretary_of_the_Interior|Secretary of the Interior]] is [[w:Doug_Burgum|Doug Burgum]], who will also be [[w:List_of_U.S._executive_branch_czars|Energy Czar]]. [[w:Department_of_Energy|Department of Energy]] [[w:United_States_Secretary_of_Energy|secretary nominee]] [[w:Chris_Wright_(energy_executive)|Chris Wright]] admits that burning fossil fuels contributes to rising temperatures, but says it poses only a modest threat to humanity, and praises it for increasing plant growth, making the planet greener, and boosting agricultural productivity. He also says that it likely reduces the annual number of temperature-related deaths. (estimates from health researchers say otherwise). He says, "It's probably almost as many positive changes as negative changes... Is it a crisis, is it the world's greatest challenge, or a big threat to the next generation? No. .. A little bit warmer isn’t a threat. If we were 5, 7, 8, 10 degrees [Celsius] warmer, that would be meaningful changes to the planet.” Scientists see a 1.5 degrees Celsius temperature as creating potentially irreversible changes for the planet, and expect to pass that mark later this year, after increasing over several decade. He criticizes the [[w:Paris_climate_agreement|Paris climate agreement]] for empowering "political actors with anti-fossil fuel agendas." Wright favors development of [[w:Geothermal_energy|geothermal energy]] and [[w:Nuclear_energy_policy_of_the_United_States|nuclear energy]], criticizing subsidies to wind and solar energy. <ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/politics/policy/who-is-chris-wright-trump-energy-secretary-9eb617dc?page=1|title=Trump’s Energy Secretary Pick Preaches the Benefits of Climate Change|last=Morenne|first=Benoit|date=9 December 2024|work=The Wall Street Journal}}&lt;nowiki&gt;</ref> === Bureau of Land Management === === Forest Service === === National Parks === == AI and Cryptocurrency Policy == [[w:David_Sacks|David Sacks]] was named "White House AI and Crypto Czar". == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Government Efficiency Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref name=":2"><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}</ref> [[w:Marc_Andreesen|Marc Andreesen]] has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. [[w:Jared_Birchall|Jared Birchall]] has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" [[w:Shaun_MacGuire|Shaun MacGuire]] has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. [[w:Antonio_Gracias|Antonio Gracias]] and [[w:Steve_Davis|Steve Davis]] from Musk's "crisis team" have been active, as has investor [[w:John_Hering|John Hering]]. Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * [[w:Mark_Pincus|Mark Pincus]] * [[w:David_Marcus|David Marcus]] * [[w:Barry_Akis|Barry Akis]] * [[w:Shervin_Pishevar|Shervin Pishevar]], who has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions."<ref name=":2" /> [[w:William_McGinley|William McGinley]] will move to a role with DOGE. Originally nominated for [[w:White_House_counsel|White House counsel]], he will be replaced in that role by [[w:David_Warrington|David Warrington]].<ref>{{Cite news|title=A White House Counsel Replaced before starting|last=Haberman|first=Maggie|date=6 December 2024|work=New York times}}</ref> The WSJ lauded without naming them, comparing them to the "dollar-a-year men" - business leaders who during WWII revolutionized industrial production to help make America the "arsenal of democracy". (WSJ, 10 December 2024) == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[w:Ministry_of_Deregulation_and_State_Transformation|Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] * [[w:Delivering_Outstanding_Government_Efficiency_Caucus|Delivering Outstanding Government Efficiency Caucus]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 1k8ouim2zwo74f3433ixzjkfdu707pe 2691732 2691731 2024-12-13T06:37:53Z Jaredscribe 2906761 /* Federal Bureau of Investigation (FBI) */ Director Christopher A. Wray announced 11 December that he would step down. Deputy Paul Abbate will be the interim director. 2691732 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission|founder=Donald Trump|extinction=4 July 2026 (planned)|mission=(In the words of president-elect Donald Trump: * dismantle government bureaucracy * slash excess regulations * cut wasteful expenditures * restructure federal agencies, * address "massive waste and fraud" in government spending}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]],<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref> with support from many [[w:Political_appointments_of_the_second_Trump_administration|Political and cabinet appointees of the second Trump administration]] and from a Congressional caucus Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == President-elect Trump has promised to "put an end to endless wars", to make [[w:NATO#NATO_defence_expenditure|NATO members pay their fair share]], end the [[w:Russian_invasion_of_Ukraine|current Russian invasion of Ukraine]], to renew the maximum-pressure policy toward Iran, and to free the hostages held in Gaza and/or ensure Israeli victory in the [[w:Israel–Hamas_war|current multi-front war launched by Iran and its proxies]]. NATO Secretary General [[w:Mark_Rutte|Mark Rutte]] publicly thanked Trump for stimulating Europe to increase national defense spending above 2%, saying "this is his doing, his success, and we need to do more, we notice."<ref>{{Cite news|url=https://www.wsj.com/video/wsj-opinion-twilight-of-the-trans-atlantic-relationship/FA4C937B-57AF-4E1D-BAC4-7293607577D1?page=1|title=WSJ Opinion: Twilight of the Trans-Atlantic Relationship|last=WSJ Opinion|date=26 November 2024|work=The Wall Street Journal}}</ref> Nominee for [[w:National_Security_Advisor|National Security Advisor]] [[w:Mike_Waltz|Mike Waltz]] To oversee the [[w:U.S._Intelligence_Community|U.S. Intelligence Community]] and NIP, and the 18 IC agencies, including the CIA, DIA, NSC, the nominee for [[w:Director_of_National_Intelligence|Director of National Intelligence]] is [[w:Tulsi_Gabbard|Tulsi Gabbard]], who is an isolationist of the [[w:Bernie_Sanders#foreign_policy|Bernie Sanders]] camp, with a long record of dogmatically opposing [[w:Foreign_policy_of_the_Trump_administration|President Trump's first term foreign policy]].<ref>{{Cite news|url=https://www.wsj.com/opinion/tulsi-gabbard-director-of-national-intelligence-donald-trump-foreign-policy-syria-israel-iran-b37aa3de|title=How Tulsi Gabbard Sees the World|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}}</ref> ''"The first act of a statesman is to recognize the type of war he is in"'', according to [[w:Carl_von_Clausewitz|Clausewitz]], given that human determination outweighs material advantages. Therefore he is advised by [[w:West_Point|West Point]] strategist [[w:John_Spencer|John Spencer]] writing in the WSJ to avoid four common foreign-policy fallacies: * the "abacus fallacy" that wars are won by superior resources, counterexample Vietnam * the "vampire fallacy" that wars are won by superior technology, counterexample Russia's failure in Ukraine, (c.f. Lt. Gen [[w:H.R._McMaster|H.R. McMaster]], 2014) * the "Zero Dark Thirty" fallacy that elevates precision strikes and special ops to the level of grand strategy or above (ibid) * and the "Peace table fallacy", which believes that all wars end in negotiation.<ref>{{Cite news|url=https://www.wsj.com/opinion/stopping-endless-wars-is-easier-said-than-done-trump-second-term-2cab9c7a?page=1|title=Stopping ‘Endless Wars’ Is Easier Said Than Done|last=Spencer|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> === Department of State === {{Main article|w:Second presidency of Donald Trump#Prospective foreign policy|w:State Department}} [[w:Marco_Rubio|Marco Rubio]] has been nominated as [[w:U.S._Secretary_of_State|U.S. Secretary of State]], overseeing $53bn and 77,880 employees ==== [[w:USAID|USAID]] ==== ==== National Endowment for Democracy ==== The [[w:National_Endowment_for_Democracy|National Endowment for Democracy]] is grant-making foundation organized as a private non-profit corporation overseen by congress, a project of Ronald Reagan announced in a [[1982 speech to British Parliament]], in which he stated that "freedom is not the sole prerogative of a lucky few, but the inalienable and universal right of all human beings", invoking the Israelites exodus and the Greeks' stand at Thermopylae. The NED is reportedly near the top of the DOGE's hit list.<ref>{{Cite news|url=https://www.wsj.com/opinion/save-a-reagan-initiative-from-the-doge-national-endowment-for-democracy-funding-2b6cc072?page=1|title=Save a Reagan Initiative From Musk and Ramaswamy|last=Galston|first=William|date=11 December 2024|work=The Wall Street Journal}}</ref> It is had a pro-freedom and [[w:Anti-communist|anti-communist]] mission to help pro-democracy leaders and groups in Asia, Africa, and Latin America, and assisted the transition of Eastern and Central European nations. The arguments being made by those in favor of defunding are that it "is a relic of the Cold War that has outlived its usefulness and no longer serves any pressing purpose in terms of advancing national interests", according to [[w:James_Piereson|James Piereson]]{{Cn}} Congress has raised its funding significanty in recent years, in a vote of confidence. === U.S. Department of Defense === U.S. DoD employees ____ civilian personel, ___ civilian contractors, and oversees a budget of _____. Nominated for [[w:Secretary_of_Defense|Secretary of Defense]] is [[w:Pete_Hegseth|Pete Hegseth]], who has been doubted by many Republican Senators{{Cn}} and supported by Trump's base.{{Cn}} The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === [[w:DARPA|DARPA]] === === US Air Force === Air Force is advancing a program called [[w:Collaborative_Combat_Aircraft|Collaborative Combat Aircraft]] to build roughly 1,000 UAVs, with [[w:Anduril|Anduril]] and [[w:General_Atomics|General Atomics]] currently building prototypes, ahead of an Air Force decision on which company or companies will be contracted to build it. The cost quickly exceeded the $2.3 billion approved for last fiscal year’s budget, according to the [[w:Congressional_Research_Service|Congressional Research Service]], prompting calls for more oversight. “''If you want to make real improvements to the defense and security of the United States of America, we would be investing more in drones, we’d be investing more in [[w:Hypersonic_weapon|hypersonic missiles]]'',” said Mr. Ramaswamy. The program for Lockheed-Martin's [[w:F-35|F-35]] stealth jet fighters, now in production, is expected to exceed $2 trillion over several decades. The Air Force on 5 December announced it would delay a decision on which company would build the next-generation crewed fighter, called [[w:Next Generation Air Dominance|Next Generation Air Dominance (NGAD)]], which was planned to replace the [[w:F-22|F-22]] and operate alongside the F-35. Mr. Musk has written that "manned fighter jets are obsolete in the age of drones.” In another post, he claimed “a reusable drone” can do everything a jet fighter can do “without all the overhead of a pilot.” Brigadier General [[w:Doug_Wickert|Doug Wickert]] said in response, “There may be some day when we can completely rely on roboticized warfare but we are a century away.... How long have we thought full self-driving was going to be on the Tesla?” <ref>{{Cite news|url=https://www.wsj.com/politics/national-security/air-force-jets-vs-drones-trump-administration-8b1620a5?page=1|title=Trump Administration Set to Decide Future of Jet Fighters|last=Seligman|first=Lara|date=6 December 2024|work=The Wall Street Journal}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Space Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. === Rail and Tunnel Authority === === Ports Authority === === Interstate Highway Authority === === [[w:Federal_Aviation_Administration|Federal Aviation Administration]] === Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} === National Air and Space Administration === The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] (NASA) had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization." In an interview Isaacman said that NASA will evolve as private space companies set their own priorities and develop technology. NASA could have a certification role for astronauts and vehicles, similar to how the Federal Aviation Administration oversees the commercial airline industry. “The FAA doesn’t build the airplanes. They don’t staff the pilots that fly you from point A to B,” he said. “That is the world that NASA is in, essentially.” He also suggested openness to new and lower cost ways of getting to the Moon and to Mars.<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} === National Oceanic and Atmospheric Administration === == Department of Education and Propaganda == [[w:United_States_Department_of_Education|Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> Nominated as [[w:US_Secretary_of_Education|Secretary of Education]] is [[w:Linda_McMahon|Linda McMahon]]. The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> During his campaign, Trump had pledged to get the "transgender insanity the hell out of schools.” Relying on the district court's decision in ''[[w:Tatel_v._Mount_Lebanon_School_District|Tatel v. Mount Lebanon School District]] , the'' attorney general and education secretary could issue a letter explaining how enforcing gender ideology violates constitutional [[w:Free_exercise_clause|First amendment right to free exercise of religion]] and the [[w:Equal_Protection_Clause|14th Amendment’s Equal Protection Clause]].<ref>{{Cite news|url=https://www.wsj.com/opinion/how-trump-can-target-transgenderism-in-schools-law-policy-education-369537a7?page=1|title=How Trump Can Target Transgenderism in Schools|last=Eden|first=Max|date=9 December 2024|work=The Wall Street Journal}}</ref> === CPB, PBS, NPR === Regarding the [[w:Corporation_for_Public_Broadcasting|Corporation for Public Broadcasting]], [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> {{As of|2024|alt=For fiscal year 2024}}, its appropriation was US$525 million, including $10 million in interest earned. The distribution of these funds was as follows:<ref>{{cite web|url=https://cpb.org/aboutcpb/financials/budget/|title=CPB Operating Budget|last=|date=2024|website=www.cpb.org|archive-url=|archive-date=|access-date=November 27, 2024|url-status=}}</ref> * $262.83M for direct grants to local public television stations; * $95.11M for television programming grants; * $81.77M for direct grants to local public radio stations; * $28.12M for the Radio National Program Production and Acquisition * $9.43M for the Radio Program Fund * $31.50 for system support * $26.25 for administration Public broadcasting stations are funded by a combination of private donations from listeners and viewers, foundations and corporations. Funding for public television comes in roughly equal parts from government (at all levels) and the private sector.<ref>{{cite web|url=http://www.cpb.org/annualreports/2013/|title=CPB 2013 Annual Report|website=www.cpb.org|archive-url=https://web.archive.org/web/20160212170045/http://cpb.org/annualreports/2013/|archive-date=February 12, 2016|access-date=May 4, 2018|url-status=dead}}</ref> == Department of Justice (DOJ) == The US. [[w:Department_of_Justice|Department of Justice]] has a 2023 budget of _____ and ___ employees. Nominated as [[w:Attorney_General|Attorney General]] is Florida AG [[w:Pam_Bondi|Pam Bondi]]{{Cn}}, after Matt Gaetz withdrew his candidacy after pressure.{{Cn}} === Federal Bureau of Investigation (FBI) === With 35,000 employees the [[w:FBI|FBI]] made a 2021 budget request for $9.8 billion. Nominated as [[w:Director_of_the_Federal_Bureau_of_Investigation|FBI Director]] is [[w:Kash_Patel|Kash Patel]], who promised to "shut down the FBI [[w:Hoover_building|Hoover building]] on day one, and open it the next day as a museum of the deep state. He said "''I would take the 7,000 employees that work in that building and send them out across the America to chase criminals"'', saying ''"Go be cops."'' He promised to retaliate against journalists and government employees who "helped Joe Biden rig the election" in 2020.<ref>{{Cite web|url=https://www.wsj.com/video/series/wsj-explains/who-is-kash-patel-donald-trumps-pick-to-lead-the-fbi/F4D38D41-013A-4B05-A170-D7394AA91C2B|title=Who Is Kash Patel, Donald Trump’s Pick to Lead the FBI?|date=6 December 2024|website=WSJ.com Video}}</ref> Director [[w:Christopher_A._Wray|Christopher A. Wray]] announced 11 December that he would step down. [[w:Deputy_Director_of_the_Federal_Bureau_of_Investigation|Deputy]] [[w:Paul_Abbate|Paul Abbate]] will be the interim director. ===== Reception and Analysis ===== His nomination "sent shock waves" through the DOJ, and his nomination has been opposed by many Republican lawmakers{{Cn}}, including former CIA director [[w:Gina_Haspell|Gina Haspell]] and AG [[w:Willliam_Barr|Willliam Barr]], who had threatened to resign if Mr. Patel were to be forced on them as a deputy, during Mr. Trump's first term.<ref name=":6">{{Cite news|url=https://www.wsj.com/opinion/kash-patel-doesnt-belong-at-the-fbi-cabinet-nominee-5ef655eb?page=1|title=Kash Patel Doesn’t Belong at the FBI: At the NSC, he was less interested in his assigned duties than in proving his loyalty to Donald Trump.|last=Bolton|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> As an author he wrote a polemical children's book lionizing "King Donald" with himself in the role of "Wizard", despite the Constitution's republican values and its prohibition on granting titles of nobility.<ref>Original synthethic statement, with constitutional context provided by <nowiki>[[User:Jaredscribe]]</nowiki></ref> He has been accused of exaggerating his roles and accomplishments, and deliberate vowing to violate the [[w:Article_Two_of_the_United_States_Constitution#Clause_5:_Caring_for_the_faithful_execution_of_the_law|"Take care" clause of Article II.3]], "''that the Laws be faithfully executed"'' by placing personal loyalties, vendettas, and hunches above his oath to the Constitution.<ref name=":6" /> He has also been accused of lying about national intelligence by [[w:Mark_Esper|Mark Esper]] in his memoir, recently again by Pence aide [[w:Olivia_Troye|Olivia Troye]], although former SoS [[w:Mike_Pompeo|Mike Pompeo]] has not yet clarified the incident in question.  He was called upon to do so in a 11 December WSJ piece by former National Security Advisor [[w:John_Bolton|John Bolton]], who also wrote, "''If illegitimate partisan prosecutions were launched [by the Biden administration], then those responsible should be held accountable in a reasoned, professional manner, not in a counter-witch hunt.  The worst response is for Mr. Trump to engage in the prosecutorial [mis]conduct he condemns [which further] politicizes and degrades the American people's faith in evenhanded law enforcement.''"<ref name=":6" /> He has received support from _____ who wrote that _______.{{Cn}} == U.S. Department of Health and Human Services (DHHS) == Nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United States Secretary of Health and Human Services]] is [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]], deputy Secretary [[w:Jim_O'Neill_(investor)|Jim O'Neill]]. Mr. Kennedy warned on 25 October that the FDA's "war on public health is about to end", accusing it of suppressing psychedelics, stem cells, raw milk, hydroxycloroquine, sunshine, and "anything else that advances human health and can't be patented by Pharma." He said that on day one he would "advise all US water systems to remove fluoride from public water".<ref>{{Cite news|title=MAHA man|date=9 November 2024|work=The Economist|department=News editors}}</ref> Nominated for [[w:US_Surgeon_General|US Surgeon General]] is [[w:Janette_Nesheiwat|Janette Nesheiwat]]. [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Nominee !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Marty_Makary|Marty Makary]] |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- | |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- | |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Dave_Weldon|Dave Weldon]] |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:Jay_Bhattacharya|Jay Bhattacharya]] |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- | |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- | |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Mehmet_Oz|Mehmet Oz]] |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- | |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- | |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === The [[w:NIH|NIH]] distributes grants of ~$50bn per year. Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]], who has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic", due to a phenomenon has been called [[Eroom’s law]], which explains that career incentives encourage “me-too research,” given that citations by other scientists “have become the dominant way to evaluate scientific contributions and scientists.” That has shifted research “toward incremental science and away from exploratory projects that are more likely to fail, but which are the fuel for future breakthroughs.”<ref name=":3">{{Cite news|url=https://www.wsj.com/opinion/jay-bhattacharya-and-the-vindication-of-the-fringe-scientists-pandemic-lockdowns-38b6aec6|title=Jay Bhattacharya and the Vindication of the ‘Fringe’ Scientists|last=Finley|first=Allysia|date=1 December 2024|work=Wall Street Journal}}</ref> Dr. Bhattacharya's February 2020 paper explaining Eroom's law, as possible explanation for slowing of pharmaceutical advances.{{Cn}} * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]], encouraging academic freedom among NIH scientists and term limits for NIH leaders. “Those kinds of reforms, I think every scientist would agree, every American would agree, it’s how you turn the NIH from something that is sort of how to control society, into something that is aimed at the discovery of truth to improve the health of Americans,” he said.<ref name=":4">{{Cite news|url=https://www.wsj.com/health/healthcare/covid-lockdown-critic-jay-bhattacharya-chosen-to-lead-nih-2958e5e2?page=1|title=Covid-Lockdown Critic Jay Bhattacharya Chosen to Lead NIH|last=Whyte|first=Liz Essley|date=26 November 2024|work=The Wall Street Journal}}</ref> * Refocusing on research on [[w:Chronic_diseases|chronic diseases]], which is underfunded, and away from [[w:Infectious_diseases|infectious diseases]], which is overfunded. * Ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague [[w:Eran_Bendavid|Eran Bendavid]], arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}&lt;nowiki&gt;</ref> Dr. Bhattacharya, [[w:Martin_Kulldorff|Martin Kulldorff]], then at Harvard, and Oxford’s [[w:Sunetra_Gupta|Sunetra Gupta]] formally expounded this idea in the [[w:Great_Barrington_Declaration|Great Barrington Declaration]] in October 2020, urging the government to focus on protecting the vulnerable while letting others go about their lives, which previous NIH director [[w:Francis_Collins|Francis Collins]] derided as "fringe science its into the political views of certain parts of our confused political establishment," and previous [[w:NIAID|NIAID]] director [[w:Chief_Medical_Advisor_to_the_President|chief medical advisor to the President]] [[w:Anthony_Fauci|Anthony Fauci]] "a quick and devastating public takedown of its premises." Some suggest the same career incentives that lead to scientific group-think in the pharmaceutical industry, also explain conformist behavior during COVID-19, due to the threat against young scientists of losing NIH funding, jobs, and career opportunities, if they were to exercise in independent judgement.<ref name=":3" /> “Dr. Jay Bhattacharya is the ideal leader to restore NIH as the international template for gold-standard science and evidence-based medicine,” DHHS Secretary nominee Kennedy wrote. "We will reform American scientific institutions so that they are worthy of trust again and will deploy the fruits of excellent science to make America healthy again!” said Dr. Bhattacharya. “Dr. Bhattacharya is a strong choice to lead the NIH,” said Dr. [[w:Ned_Sharpless|Ned Sharpless]], a former [[w:National_Cancer_Institute|National Cancer Institute]] director. “The support of moderate Senate Republicans will be critical to NIH funding, and Dr. Bhattacharya’s Covid work will give him credibility with this constituency.”<ref name=":4" /> === Food and Drug Administration === The FDA in 2022 had 18,000 employees<ref name="fy2022">{{cite web|url=https://www.fda.gov/media/149613/download|title=FY 2022 FDA Budget Request|publisher=FDA|archive-url=https://web.archive.org/web/20230602090805/https://www.fda.gov/media/149613/download|archive-date=June 2, 2023|access-date=January 14, 2022|url-status=live}}</ref> and a budget of $6.5{{nbsp}}billion (2022)<ref name="fy2022" /> Nominated as director is [[w:Marty_Makary|Marty Makary]]. == Department of Agriculture and Food == [[w:Department_of_Agriculture|Department of Agriculture]] (USDA) had 2023 budget of ___ and ____ employees. Nominated for [[w:Secretary_of_Agriculture|Secretary of Agriculture]] is [[w:Brooke_Rollins|Brooke Rollins]], who had earlier served on the [[w:Office_of_American_Innovation|Office of American Innovation]] under [[w:Jared_Kushner|Jared Kushner]], and served as director of [[w:Domestic_Policy_Council|Domestic Policy Council]]. She has not endorsed the "[[w:Make_America_Healthy_Again|Make America Healthy Again]]" agenda of RFK Jr. (and his colleagues Jay Bhattacharcya and others) who promised to "reverse 80 years of farm policy" and complains of the $30 billion/year farms subsidies. Kennedy wants to remove soda from food aid, and ultra-processed food from both [[w:Food-stamps|food-stamp]] benefits and [[w:School_meals|school meals]], both of which are overseen by the USDA; an effort that in the past has been opposed by the food industry, lawmakers, and some anti-hunger advocacy groups.<ref name=":5" /> RFK Jr.'s team had recommended [[w:Sid_Miller|Sid Miller]] for the role, and a group of farmers he had asked to vet candidates had proposed [[w:John_Kempf|John Kempf]].<ref name=":5">{{Cite news|url=https://www.wsj.com/politics/policy/trump-agriculture-pick-brooke-rollins-rfk-jr-1a85beda?page=1|title=RFK Jr. Team Skeptical About USDA Pick|last=Andrews|first=Natalie|date=11 December 2024|work=The Wall Street Journal|others=et al}}</ref> He has also called for re-examining the the [[standards regulating the use of pesticides]], especially [[w:Glyphosate|glyphosate]], the world's most widely used [[w:Herbicide|herbicide]] and the active ingredient in [[w:Roundup|Roundup]], used as a weedkiller in major [[w:U.S._commodity_crops|U.S. commodity crops]].[[w:Herbicide|herbicide]]<ref name=":5" /> === Food Stamps === === School Meals === === Farm Subsidies === == Department of Treasury and Reserve == [[w:Departments_of_Treasury|Departments of Treasury]] has 2023 budget of ____ and ___ employees. Nominated for [[w:Secretary_of_the_Treasury|Secretary of the Treasury]] is [[w:Scott_Bessent|Scott Bessent]]. === Consumer Financial Protection Bureau === The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> === Securities and Exchange Commission === [[w:Securities_and_Exchange_Commission|Securities and Exchange Commission]] === Internal Revenue Service === [[w:Internal_Revenue_Service|Internal Revenue Service]] === Federal Reserve === Mr. Musk has suggested starting a "[[w:Sovereign_Wealth_Fund|Sovereign Wealth Fund]]" like Texas and other U.S. states, instead of hosting a [[w:National_debt|national debt]]. Ron Paul and others have called for abolishing America's [[w:Central_Bank|Central Bank]], the [[w:Federal_Reserve|Federal Reserve System]], which Mr. Musk appeared to endorse. === American Sovereign Wealth Fund === == Department of Industry, Labor, and Commerce == [[w:Department_of_Commerce|Department of Commerce]] has 2023 budget of _____ and _____ employees. Nominated as [[w:Secretary_of_Commerce|Secretary of Commerce]] is [[w:Howard_Lutnick|Howard Lutnick]] [[w:Department_of_Labor|Department of Labor]] has 2023 budget of _____ and ____ employees. Nominated as [[w:Secretary_of_Labor|Secretary of Labor]] is [[w:Lori_Chavez-Remer|Lori Chavez-Remer]] == Departments of Energy and Interior == Nominee for [[w:US_Secretary_of_the_Interior|Secretary of the Interior]] is [[w:Doug_Burgum|Doug Burgum]], who will also be [[w:List_of_U.S._executive_branch_czars|Energy Czar]]. [[w:Department_of_Energy|Department of Energy]] [[w:United_States_Secretary_of_Energy|secretary nominee]] [[w:Chris_Wright_(energy_executive)|Chris Wright]] admits that burning fossil fuels contributes to rising temperatures, but says it poses only a modest threat to humanity, and praises it for increasing plant growth, making the planet greener, and boosting agricultural productivity. He also says that it likely reduces the annual number of temperature-related deaths. (estimates from health researchers say otherwise). He says, "It's probably almost as many positive changes as negative changes... Is it a crisis, is it the world's greatest challenge, or a big threat to the next generation? No. .. A little bit warmer isn’t a threat. If we were 5, 7, 8, 10 degrees [Celsius] warmer, that would be meaningful changes to the planet.” Scientists see a 1.5 degrees Celsius temperature as creating potentially irreversible changes for the planet, and expect to pass that mark later this year, after increasing over several decade. He criticizes the [[w:Paris_climate_agreement|Paris climate agreement]] for empowering "political actors with anti-fossil fuel agendas." Wright favors development of [[w:Geothermal_energy|geothermal energy]] and [[w:Nuclear_energy_policy_of_the_United_States|nuclear energy]], criticizing subsidies to wind and solar energy. <ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/politics/policy/who-is-chris-wright-trump-energy-secretary-9eb617dc?page=1|title=Trump’s Energy Secretary Pick Preaches the Benefits of Climate Change|last=Morenne|first=Benoit|date=9 December 2024|work=The Wall Street Journal}}&lt;nowiki&gt;</ref> === Bureau of Land Management === === Forest Service === === National Parks === == AI and Cryptocurrency Policy == [[w:David_Sacks|David Sacks]] was named "White House AI and Crypto Czar". == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Government Efficiency Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref name=":2"><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}</ref> [[w:Marc_Andreesen|Marc Andreesen]] has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. [[w:Jared_Birchall|Jared Birchall]] has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" [[w:Shaun_MacGuire|Shaun MacGuire]] has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. [[w:Antonio_Gracias|Antonio Gracias]] and [[w:Steve_Davis|Steve Davis]] from Musk's "crisis team" have been active, as has investor [[w:John_Hering|John Hering]]. Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * [[w:Mark_Pincus|Mark Pincus]] * [[w:David_Marcus|David Marcus]] * [[w:Barry_Akis|Barry Akis]] * [[w:Shervin_Pishevar|Shervin Pishevar]], who has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions."<ref name=":2" /> [[w:William_McGinley|William McGinley]] will move to a role with DOGE. Originally nominated for [[w:White_House_counsel|White House counsel]], he will be replaced in that role by [[w:David_Warrington|David Warrington]].<ref>{{Cite news|title=A White House Counsel Replaced before starting|last=Haberman|first=Maggie|date=6 December 2024|work=New York times}}</ref> The WSJ lauded without naming them, comparing them to the "dollar-a-year men" - business leaders who during WWII revolutionized industrial production to help make America the "arsenal of democracy". (WSJ, 10 December 2024) == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[w:Ministry_of_Deregulation_and_State_Transformation|Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] * [[w:Delivering_Outstanding_Government_Efficiency_Caucus|Delivering Outstanding Government Efficiency Caucus]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} e2aypihzptmc16l4pxl1ef00hyypp6z 2691734 2691732 2024-12-13T06:42:27Z Jaredscribe 2906761 /* Federal Bureau of Investigation (FBI) */ In my view, this is the best way to avoid dragging the Bureau deeper into the fray." Mr. Trump said "the resignation of Christopher Wray is a great day for America". 2691734 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission|founder=Donald Trump|extinction=4 July 2026 (planned)|mission=(In the words of president-elect Donald Trump: * dismantle government bureaucracy * slash excess regulations * cut wasteful expenditures * restructure federal agencies, * address "massive waste and fraud" in government spending}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]],<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref> with support from many [[w:Political_appointments_of_the_second_Trump_administration|Political and cabinet appointees of the second Trump administration]] and from a Congressional caucus Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == President-elect Trump has promised to "put an end to endless wars", to make [[w:NATO#NATO_defence_expenditure|NATO members pay their fair share]], end the [[w:Russian_invasion_of_Ukraine|current Russian invasion of Ukraine]], to renew the maximum-pressure policy toward Iran, and to free the hostages held in Gaza and/or ensure Israeli victory in the [[w:Israel–Hamas_war|current multi-front war launched by Iran and its proxies]]. NATO Secretary General [[w:Mark_Rutte|Mark Rutte]] publicly thanked Trump for stimulating Europe to increase national defense spending above 2%, saying "this is his doing, his success, and we need to do more, we notice."<ref>{{Cite news|url=https://www.wsj.com/video/wsj-opinion-twilight-of-the-trans-atlantic-relationship/FA4C937B-57AF-4E1D-BAC4-7293607577D1?page=1|title=WSJ Opinion: Twilight of the Trans-Atlantic Relationship|last=WSJ Opinion|date=26 November 2024|work=The Wall Street Journal}}</ref> Nominee for [[w:National_Security_Advisor|National Security Advisor]] [[w:Mike_Waltz|Mike Waltz]] To oversee the [[w:U.S._Intelligence_Community|U.S. Intelligence Community]] and NIP, and the 18 IC agencies, including the CIA, DIA, NSC, the nominee for [[w:Director_of_National_Intelligence|Director of National Intelligence]] is [[w:Tulsi_Gabbard|Tulsi Gabbard]], who is an isolationist of the [[w:Bernie_Sanders#foreign_policy|Bernie Sanders]] camp, with a long record of dogmatically opposing [[w:Foreign_policy_of_the_Trump_administration|President Trump's first term foreign policy]].<ref>{{Cite news|url=https://www.wsj.com/opinion/tulsi-gabbard-director-of-national-intelligence-donald-trump-foreign-policy-syria-israel-iran-b37aa3de|title=How Tulsi Gabbard Sees the World|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}}</ref> ''"The first act of a statesman is to recognize the type of war he is in"'', according to [[w:Carl_von_Clausewitz|Clausewitz]], given that human determination outweighs material advantages. Therefore he is advised by [[w:West_Point|West Point]] strategist [[w:John_Spencer|John Spencer]] writing in the WSJ to avoid four common foreign-policy fallacies: * the "abacus fallacy" that wars are won by superior resources, counterexample Vietnam * the "vampire fallacy" that wars are won by superior technology, counterexample Russia's failure in Ukraine, (c.f. Lt. Gen [[w:H.R._McMaster|H.R. McMaster]], 2014) * the "Zero Dark Thirty" fallacy that elevates precision strikes and special ops to the level of grand strategy or above (ibid) * and the "Peace table fallacy", which believes that all wars end in negotiation.<ref>{{Cite news|url=https://www.wsj.com/opinion/stopping-endless-wars-is-easier-said-than-done-trump-second-term-2cab9c7a?page=1|title=Stopping ‘Endless Wars’ Is Easier Said Than Done|last=Spencer|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> === Department of State === {{Main article|w:Second presidency of Donald Trump#Prospective foreign policy|w:State Department}} [[w:Marco_Rubio|Marco Rubio]] has been nominated as [[w:U.S._Secretary_of_State|U.S. Secretary of State]], overseeing $53bn and 77,880 employees ==== [[w:USAID|USAID]] ==== ==== National Endowment for Democracy ==== The [[w:National_Endowment_for_Democracy|National Endowment for Democracy]] is grant-making foundation organized as a private non-profit corporation overseen by congress, a project of Ronald Reagan announced in a [[1982 speech to British Parliament]], in which he stated that "freedom is not the sole prerogative of a lucky few, but the inalienable and universal right of all human beings", invoking the Israelites exodus and the Greeks' stand at Thermopylae. The NED is reportedly near the top of the DOGE's hit list.<ref>{{Cite news|url=https://www.wsj.com/opinion/save-a-reagan-initiative-from-the-doge-national-endowment-for-democracy-funding-2b6cc072?page=1|title=Save a Reagan Initiative From Musk and Ramaswamy|last=Galston|first=William|date=11 December 2024|work=The Wall Street Journal}}</ref> It is had a pro-freedom and [[w:Anti-communist|anti-communist]] mission to help pro-democracy leaders and groups in Asia, Africa, and Latin America, and assisted the transition of Eastern and Central European nations. The arguments being made by those in favor of defunding are that it "is a relic of the Cold War that has outlived its usefulness and no longer serves any pressing purpose in terms of advancing national interests", according to [[w:James_Piereson|James Piereson]]{{Cn}} Congress has raised its funding significanty in recent years, in a vote of confidence. === U.S. Department of Defense === U.S. DoD employees ____ civilian personel, ___ civilian contractors, and oversees a budget of _____. Nominated for [[w:Secretary_of_Defense|Secretary of Defense]] is [[w:Pete_Hegseth|Pete Hegseth]], who has been doubted by many Republican Senators{{Cn}} and supported by Trump's base.{{Cn}} The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === [[w:DARPA|DARPA]] === === US Air Force === Air Force is advancing a program called [[w:Collaborative_Combat_Aircraft|Collaborative Combat Aircraft]] to build roughly 1,000 UAVs, with [[w:Anduril|Anduril]] and [[w:General_Atomics|General Atomics]] currently building prototypes, ahead of an Air Force decision on which company or companies will be contracted to build it. The cost quickly exceeded the $2.3 billion approved for last fiscal year’s budget, according to the [[w:Congressional_Research_Service|Congressional Research Service]], prompting calls for more oversight. “''If you want to make real improvements to the defense and security of the United States of America, we would be investing more in drones, we’d be investing more in [[w:Hypersonic_weapon|hypersonic missiles]]'',” said Mr. Ramaswamy. The program for Lockheed-Martin's [[w:F-35|F-35]] stealth jet fighters, now in production, is expected to exceed $2 trillion over several decades. The Air Force on 5 December announced it would delay a decision on which company would build the next-generation crewed fighter, called [[w:Next Generation Air Dominance|Next Generation Air Dominance (NGAD)]], which was planned to replace the [[w:F-22|F-22]] and operate alongside the F-35. Mr. Musk has written that "manned fighter jets are obsolete in the age of drones.” In another post, he claimed “a reusable drone” can do everything a jet fighter can do “without all the overhead of a pilot.” Brigadier General [[w:Doug_Wickert|Doug Wickert]] said in response, “There may be some day when we can completely rely on roboticized warfare but we are a century away.... How long have we thought full self-driving was going to be on the Tesla?” <ref>{{Cite news|url=https://www.wsj.com/politics/national-security/air-force-jets-vs-drones-trump-administration-8b1620a5?page=1|title=Trump Administration Set to Decide Future of Jet Fighters|last=Seligman|first=Lara|date=6 December 2024|work=The Wall Street Journal}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Space Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. === Rail and Tunnel Authority === === Ports Authority === === Interstate Highway Authority === === [[w:Federal_Aviation_Administration|Federal Aviation Administration]] === Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} === National Air and Space Administration === The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] (NASA) had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization." In an interview Isaacman said that NASA will evolve as private space companies set their own priorities and develop technology. NASA could have a certification role for astronauts and vehicles, similar to how the Federal Aviation Administration oversees the commercial airline industry. “The FAA doesn’t build the airplanes. They don’t staff the pilots that fly you from point A to B,” he said. “That is the world that NASA is in, essentially.” He also suggested openness to new and lower cost ways of getting to the Moon and to Mars.<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} === National Oceanic and Atmospheric Administration === == Department of Education and Propaganda == [[w:United_States_Department_of_Education|Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> Nominated as [[w:US_Secretary_of_Education|Secretary of Education]] is [[w:Linda_McMahon|Linda McMahon]]. The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> During his campaign, Trump had pledged to get the "transgender insanity the hell out of schools.” Relying on the district court's decision in ''[[w:Tatel_v._Mount_Lebanon_School_District|Tatel v. Mount Lebanon School District]] , the'' attorney general and education secretary could issue a letter explaining how enforcing gender ideology violates constitutional [[w:Free_exercise_clause|First amendment right to free exercise of religion]] and the [[w:Equal_Protection_Clause|14th Amendment’s Equal Protection Clause]].<ref>{{Cite news|url=https://www.wsj.com/opinion/how-trump-can-target-transgenderism-in-schools-law-policy-education-369537a7?page=1|title=How Trump Can Target Transgenderism in Schools|last=Eden|first=Max|date=9 December 2024|work=The Wall Street Journal}}</ref> === CPB, PBS, NPR === Regarding the [[w:Corporation_for_Public_Broadcasting|Corporation for Public Broadcasting]], [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> {{As of|2024|alt=For fiscal year 2024}}, its appropriation was US$525 million, including $10 million in interest earned. The distribution of these funds was as follows:<ref>{{cite web|url=https://cpb.org/aboutcpb/financials/budget/|title=CPB Operating Budget|last=|date=2024|website=www.cpb.org|archive-url=|archive-date=|access-date=November 27, 2024|url-status=}}</ref> * $262.83M for direct grants to local public television stations; * $95.11M for television programming grants; * $81.77M for direct grants to local public radio stations; * $28.12M for the Radio National Program Production and Acquisition * $9.43M for the Radio Program Fund * $31.50 for system support * $26.25 for administration Public broadcasting stations are funded by a combination of private donations from listeners and viewers, foundations and corporations. Funding for public television comes in roughly equal parts from government (at all levels) and the private sector.<ref>{{cite web|url=http://www.cpb.org/annualreports/2013/|title=CPB 2013 Annual Report|website=www.cpb.org|archive-url=https://web.archive.org/web/20160212170045/http://cpb.org/annualreports/2013/|archive-date=February 12, 2016|access-date=May 4, 2018|url-status=dead}}</ref> == Department of Justice (DOJ) == The US. [[w:Department_of_Justice|Department of Justice]] has a 2023 budget of _____ and ___ employees. Nominated as [[w:Attorney_General|Attorney General]] is Florida AG [[w:Pam_Bondi|Pam Bondi]]{{Cn}}, after Matt Gaetz withdrew his candidacy after pressure.{{Cn}} === Federal Bureau of Investigation (FBI) === With 35,000 employees the [[w:FBI|FBI]] made a 2021 budget request for $9.8 billion. Nominated as [[w:Director_of_the_Federal_Bureau_of_Investigation|FBI Director]] is [[w:Kash_Patel|Kash Patel]], who promised to "shut down the FBI [[w:Hoover_building|Hoover building]] on day one, and open it the next day as a museum of the deep state. He said "''I would take the 7,000 employees that work in that building and send them out across the America to chase criminals"'', saying ''"Go be cops."'' He promised to retaliate against journalists and government employees who "helped Joe Biden rig the election" in 2020.<ref>{{Cite web|url=https://www.wsj.com/video/series/wsj-explains/who-is-kash-patel-donald-trumps-pick-to-lead-the-fbi/F4D38D41-013A-4B05-A170-D7394AA91C2B|title=Who Is Kash Patel, Donald Trump’s Pick to Lead the FBI?|date=6 December 2024|website=WSJ.com Video}}</ref> Director [[w:Christopher_A._Wray|Christopher A. Wray]] announced 11 December that he would step down. [[w:Deputy_Director_of_the_Federal_Bureau_of_Investigation|Deputy]] [[w:Paul_Abbate|Paul Abbate]] will be the interim director, saying “In my view, this is the best way to avoid dragging the Bureau deeper into the fray." Mr. Trump said "the resignation of Christopher Wray is a great day for America".<ref>{{Cite news|url=https://www.wsj.com/politics/policy/christopher-wray-fbi-director-resigns-69069f42?page=1|title=Christopher Wray to Step Down as FBI Director|last=Gurman|first=Sadie|date=11 December 2024|work=The Wall Street Journal}}</ref> ===== Reception and Analysis ===== His nomination "sent shock waves" through the DOJ, and his nomination has been opposed by many Republican lawmakers{{Cn}}, including former CIA director [[w:Gina_Haspell|Gina Haspell]] and AG [[w:Willliam_Barr|Willliam Barr]], who had threatened to resign if Mr. Patel were to be forced on them as a deputy, during Mr. Trump's first term.<ref name=":6">{{Cite news|url=https://www.wsj.com/opinion/kash-patel-doesnt-belong-at-the-fbi-cabinet-nominee-5ef655eb?page=1|title=Kash Patel Doesn’t Belong at the FBI: At the NSC, he was less interested in his assigned duties than in proving his loyalty to Donald Trump.|last=Bolton|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> As an author he wrote a polemical children's book lionizing "King Donald" with himself in the role of "Wizard", despite the Constitution's republican values and its prohibition on granting titles of nobility.<ref>Original synthethic statement, with constitutional context provided by <nowiki>[[User:Jaredscribe]]</nowiki></ref> He has been accused of exaggerating his roles and accomplishments, and deliberate vowing to violate the [[w:Article_Two_of_the_United_States_Constitution#Clause_5:_Caring_for_the_faithful_execution_of_the_law|"Take care" clause of Article II.3]], "''that the Laws be faithfully executed"'' by placing personal loyalties, vendettas, and hunches above his oath to the Constitution.<ref name=":6" /> He has also been accused of lying about national intelligence by [[w:Mark_Esper|Mark Esper]] in his memoir, recently again by Pence aide [[w:Olivia_Troye|Olivia Troye]], although former SoS [[w:Mike_Pompeo|Mike Pompeo]] has not yet clarified the incident in question.  He was called upon to do so in a 11 December WSJ piece by former National Security Advisor [[w:John_Bolton|John Bolton]], who also wrote, "''If illegitimate partisan prosecutions were launched [by the Biden administration], then those responsible should be held accountable in a reasoned, professional manner, not in a counter-witch hunt.  The worst response is for Mr. Trump to engage in the prosecutorial [mis]conduct he condemns [which further] politicizes and degrades the American people's faith in evenhanded law enforcement.''"<ref name=":6" /> He has received support from _____ who wrote that _______.{{Cn}} == U.S. Department of Health and Human Services (DHHS) == Nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United States Secretary of Health and Human Services]] is [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]], deputy Secretary [[w:Jim_O'Neill_(investor)|Jim O'Neill]]. Mr. Kennedy warned on 25 October that the FDA's "war on public health is about to end", accusing it of suppressing psychedelics, stem cells, raw milk, hydroxycloroquine, sunshine, and "anything else that advances human health and can't be patented by Pharma." He said that on day one he would "advise all US water systems to remove fluoride from public water".<ref>{{Cite news|title=MAHA man|date=9 November 2024|work=The Economist|department=News editors}}</ref> Nominated for [[w:US_Surgeon_General|US Surgeon General]] is [[w:Janette_Nesheiwat|Janette Nesheiwat]]. [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Nominee !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Marty_Makary|Marty Makary]] |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- | |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- | |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Dave_Weldon|Dave Weldon]] |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:Jay_Bhattacharya|Jay Bhattacharya]] |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- | |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- | |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Mehmet_Oz|Mehmet Oz]] |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- | |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- | |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === The [[w:NIH|NIH]] distributes grants of ~$50bn per year. Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]], who has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic", due to a phenomenon has been called [[Eroom’s law]], which explains that career incentives encourage “me-too research,” given that citations by other scientists “have become the dominant way to evaluate scientific contributions and scientists.” That has shifted research “toward incremental science and away from exploratory projects that are more likely to fail, but which are the fuel for future breakthroughs.”<ref name=":3">{{Cite news|url=https://www.wsj.com/opinion/jay-bhattacharya-and-the-vindication-of-the-fringe-scientists-pandemic-lockdowns-38b6aec6|title=Jay Bhattacharya and the Vindication of the ‘Fringe’ Scientists|last=Finley|first=Allysia|date=1 December 2024|work=Wall Street Journal}}</ref> Dr. Bhattacharya's February 2020 paper explaining Eroom's law, as possible explanation for slowing of pharmaceutical advances.{{Cn}} * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]], encouraging academic freedom among NIH scientists and term limits for NIH leaders. “Those kinds of reforms, I think every scientist would agree, every American would agree, it’s how you turn the NIH from something that is sort of how to control society, into something that is aimed at the discovery of truth to improve the health of Americans,” he said.<ref name=":4">{{Cite news|url=https://www.wsj.com/health/healthcare/covid-lockdown-critic-jay-bhattacharya-chosen-to-lead-nih-2958e5e2?page=1|title=Covid-Lockdown Critic Jay Bhattacharya Chosen to Lead NIH|last=Whyte|first=Liz Essley|date=26 November 2024|work=The Wall Street Journal}}</ref> * Refocusing on research on [[w:Chronic_diseases|chronic diseases]], which is underfunded, and away from [[w:Infectious_diseases|infectious diseases]], which is overfunded. * Ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague [[w:Eran_Bendavid|Eran Bendavid]], arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}&lt;nowiki&gt;</ref> Dr. Bhattacharya, [[w:Martin_Kulldorff|Martin Kulldorff]], then at Harvard, and Oxford’s [[w:Sunetra_Gupta|Sunetra Gupta]] formally expounded this idea in the [[w:Great_Barrington_Declaration|Great Barrington Declaration]] in October 2020, urging the government to focus on protecting the vulnerable while letting others go about their lives, which previous NIH director [[w:Francis_Collins|Francis Collins]] derided as "fringe science its into the political views of certain parts of our confused political establishment," and previous [[w:NIAID|NIAID]] director [[w:Chief_Medical_Advisor_to_the_President|chief medical advisor to the President]] [[w:Anthony_Fauci|Anthony Fauci]] "a quick and devastating public takedown of its premises." Some suggest the same career incentives that lead to scientific group-think in the pharmaceutical industry, also explain conformist behavior during COVID-19, due to the threat against young scientists of losing NIH funding, jobs, and career opportunities, if they were to exercise in independent judgement.<ref name=":3" /> “Dr. Jay Bhattacharya is the ideal leader to restore NIH as the international template for gold-standard science and evidence-based medicine,” DHHS Secretary nominee Kennedy wrote. "We will reform American scientific institutions so that they are worthy of trust again and will deploy the fruits of excellent science to make America healthy again!” said Dr. Bhattacharya. “Dr. Bhattacharya is a strong choice to lead the NIH,” said Dr. [[w:Ned_Sharpless|Ned Sharpless]], a former [[w:National_Cancer_Institute|National Cancer Institute]] director. “The support of moderate Senate Republicans will be critical to NIH funding, and Dr. Bhattacharya’s Covid work will give him credibility with this constituency.”<ref name=":4" /> === Food and Drug Administration === The FDA in 2022 had 18,000 employees<ref name="fy2022">{{cite web|url=https://www.fda.gov/media/149613/download|title=FY 2022 FDA Budget Request|publisher=FDA|archive-url=https://web.archive.org/web/20230602090805/https://www.fda.gov/media/149613/download|archive-date=June 2, 2023|access-date=January 14, 2022|url-status=live}}</ref> and a budget of $6.5{{nbsp}}billion (2022)<ref name="fy2022" /> Nominated as director is [[w:Marty_Makary|Marty Makary]]. == Department of Agriculture and Food == [[w:Department_of_Agriculture|Department of Agriculture]] (USDA) had 2023 budget of ___ and ____ employees. Nominated for [[w:Secretary_of_Agriculture|Secretary of Agriculture]] is [[w:Brooke_Rollins|Brooke Rollins]], who had earlier served on the [[w:Office_of_American_Innovation|Office of American Innovation]] under [[w:Jared_Kushner|Jared Kushner]], and served as director of [[w:Domestic_Policy_Council|Domestic Policy Council]]. She has not endorsed the "[[w:Make_America_Healthy_Again|Make America Healthy Again]]" agenda of RFK Jr. (and his colleagues Jay Bhattacharcya and others) who promised to "reverse 80 years of farm policy" and complains of the $30 billion/year farms subsidies. Kennedy wants to remove soda from food aid, and ultra-processed food from both [[w:Food-stamps|food-stamp]] benefits and [[w:School_meals|school meals]], both of which are overseen by the USDA; an effort that in the past has been opposed by the food industry, lawmakers, and some anti-hunger advocacy groups.<ref name=":5" /> RFK Jr.'s team had recommended [[w:Sid_Miller|Sid Miller]] for the role, and a group of farmers he had asked to vet candidates had proposed [[w:John_Kempf|John Kempf]].<ref name=":5">{{Cite news|url=https://www.wsj.com/politics/policy/trump-agriculture-pick-brooke-rollins-rfk-jr-1a85beda?page=1|title=RFK Jr. Team Skeptical About USDA Pick|last=Andrews|first=Natalie|date=11 December 2024|work=The Wall Street Journal|others=et al}}</ref> He has also called for re-examining the the [[standards regulating the use of pesticides]], especially [[w:Glyphosate|glyphosate]], the world's most widely used [[w:Herbicide|herbicide]] and the active ingredient in [[w:Roundup|Roundup]], used as a weedkiller in major [[w:U.S._commodity_crops|U.S. commodity crops]].[[w:Herbicide|herbicide]]<ref name=":5" /> === Food Stamps === === School Meals === === Farm Subsidies === == Department of Treasury and Reserve == [[w:Departments_of_Treasury|Departments of Treasury]] has 2023 budget of ____ and ___ employees. Nominated for [[w:Secretary_of_the_Treasury|Secretary of the Treasury]] is [[w:Scott_Bessent|Scott Bessent]]. === Consumer Financial Protection Bureau === The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> === Securities and Exchange Commission === [[w:Securities_and_Exchange_Commission|Securities and Exchange Commission]] === Internal Revenue Service === [[w:Internal_Revenue_Service|Internal Revenue Service]] === Federal Reserve === Mr. Musk has suggested starting a "[[w:Sovereign_Wealth_Fund|Sovereign Wealth Fund]]" like Texas and other U.S. states, instead of hosting a [[w:National_debt|national debt]]. Ron Paul and others have called for abolishing America's [[w:Central_Bank|Central Bank]], the [[w:Federal_Reserve|Federal Reserve System]], which Mr. Musk appeared to endorse. === American Sovereign Wealth Fund === == Department of Industry, Labor, and Commerce == [[w:Department_of_Commerce|Department of Commerce]] has 2023 budget of _____ and _____ employees. Nominated as [[w:Secretary_of_Commerce|Secretary of Commerce]] is [[w:Howard_Lutnick|Howard Lutnick]] [[w:Department_of_Labor|Department of Labor]] has 2023 budget of _____ and ____ employees. Nominated as [[w:Secretary_of_Labor|Secretary of Labor]] is [[w:Lori_Chavez-Remer|Lori Chavez-Remer]] == Departments of Energy and Interior == Nominee for [[w:US_Secretary_of_the_Interior|Secretary of the Interior]] is [[w:Doug_Burgum|Doug Burgum]], who will also be [[w:List_of_U.S._executive_branch_czars|Energy Czar]]. [[w:Department_of_Energy|Department of Energy]] [[w:United_States_Secretary_of_Energy|secretary nominee]] [[w:Chris_Wright_(energy_executive)|Chris Wright]] admits that burning fossil fuels contributes to rising temperatures, but says it poses only a modest threat to humanity, and praises it for increasing plant growth, making the planet greener, and boosting agricultural productivity. He also says that it likely reduces the annual number of temperature-related deaths. (estimates from health researchers say otherwise). He says, "It's probably almost as many positive changes as negative changes... Is it a crisis, is it the world's greatest challenge, or a big threat to the next generation? No. .. A little bit warmer isn’t a threat. If we were 5, 7, 8, 10 degrees [Celsius] warmer, that would be meaningful changes to the planet.” Scientists see a 1.5 degrees Celsius temperature as creating potentially irreversible changes for the planet, and expect to pass that mark later this year, after increasing over several decade. He criticizes the [[w:Paris_climate_agreement|Paris climate agreement]] for empowering "political actors with anti-fossil fuel agendas." Wright favors development of [[w:Geothermal_energy|geothermal energy]] and [[w:Nuclear_energy_policy_of_the_United_States|nuclear energy]], criticizing subsidies to wind and solar energy. <ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/politics/policy/who-is-chris-wright-trump-energy-secretary-9eb617dc?page=1|title=Trump’s Energy Secretary Pick Preaches the Benefits of Climate Change|last=Morenne|first=Benoit|date=9 December 2024|work=The Wall Street Journal}}&lt;nowiki&gt;</ref> === Bureau of Land Management === === Forest Service === === National Parks === == AI and Cryptocurrency Policy == [[w:David_Sacks|David Sacks]] was named "White House AI and Crypto Czar". == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Government Efficiency Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref name=":2"><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}</ref> [[w:Marc_Andreesen|Marc Andreesen]] has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. [[w:Jared_Birchall|Jared Birchall]] has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" [[w:Shaun_MacGuire|Shaun MacGuire]] has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. [[w:Antonio_Gracias|Antonio Gracias]] and [[w:Steve_Davis|Steve Davis]] from Musk's "crisis team" have been active, as has investor [[w:John_Hering|John Hering]]. Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * [[w:Mark_Pincus|Mark Pincus]] * [[w:David_Marcus|David Marcus]] * [[w:Barry_Akis|Barry Akis]] * [[w:Shervin_Pishevar|Shervin Pishevar]], who has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions."<ref name=":2" /> [[w:William_McGinley|William McGinley]] will move to a role with DOGE. Originally nominated for [[w:White_House_counsel|White House counsel]], he will be replaced in that role by [[w:David_Warrington|David Warrington]].<ref>{{Cite news|title=A White House Counsel Replaced before starting|last=Haberman|first=Maggie|date=6 December 2024|work=New York times}}</ref> The WSJ lauded without naming them, comparing them to the "dollar-a-year men" - business leaders who during WWII revolutionized industrial production to help make America the "arsenal of democracy". (WSJ, 10 December 2024) == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[w:Ministry_of_Deregulation_and_State_Transformation|Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] * [[w:Delivering_Outstanding_Government_Efficiency_Caucus|Delivering Outstanding Government Efficiency Caucus]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} rjagdulnv1cr81k6zau8fgu5ued8ddz 2691743 2691734 2024-12-13T07:28:19Z Jaredscribe 2906761 /* Reception and Analysis */ [[w:Foreign_Emoluments_Clause]] 2691743 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission|founder=Donald Trump|extinction=4 July 2026 (planned)|mission=(In the words of president-elect Donald Trump: * dismantle government bureaucracy * slash excess regulations * cut wasteful expenditures * restructure federal agencies, * address "massive waste and fraud" in government spending}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]],<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref> with support from many [[w:Political_appointments_of_the_second_Trump_administration|Political and cabinet appointees of the second Trump administration]] and from a Congressional caucus Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == President-elect Trump has promised to "put an end to endless wars", to make [[w:NATO#NATO_defence_expenditure|NATO members pay their fair share]], end the [[w:Russian_invasion_of_Ukraine|current Russian invasion of Ukraine]], to renew the maximum-pressure policy toward Iran, and to free the hostages held in Gaza and/or ensure Israeli victory in the [[w:Israel–Hamas_war|current multi-front war launched by Iran and its proxies]]. NATO Secretary General [[w:Mark_Rutte|Mark Rutte]] publicly thanked Trump for stimulating Europe to increase national defense spending above 2%, saying "this is his doing, his success, and we need to do more, we notice."<ref>{{Cite news|url=https://www.wsj.com/video/wsj-opinion-twilight-of-the-trans-atlantic-relationship/FA4C937B-57AF-4E1D-BAC4-7293607577D1?page=1|title=WSJ Opinion: Twilight of the Trans-Atlantic Relationship|last=WSJ Opinion|date=26 November 2024|work=The Wall Street Journal}}</ref> Nominee for [[w:National_Security_Advisor|National Security Advisor]] [[w:Mike_Waltz|Mike Waltz]] To oversee the [[w:U.S._Intelligence_Community|U.S. Intelligence Community]] and NIP, and the 18 IC agencies, including the CIA, DIA, NSC, the nominee for [[w:Director_of_National_Intelligence|Director of National Intelligence]] is [[w:Tulsi_Gabbard|Tulsi Gabbard]], who is an isolationist of the [[w:Bernie_Sanders#foreign_policy|Bernie Sanders]] camp, with a long record of dogmatically opposing [[w:Foreign_policy_of_the_Trump_administration|President Trump's first term foreign policy]].<ref>{{Cite news|url=https://www.wsj.com/opinion/tulsi-gabbard-director-of-national-intelligence-donald-trump-foreign-policy-syria-israel-iran-b37aa3de|title=How Tulsi Gabbard Sees the World|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}}</ref> ''"The first act of a statesman is to recognize the type of war he is in"'', according to [[w:Carl_von_Clausewitz|Clausewitz]], given that human determination outweighs material advantages. Therefore he is advised by [[w:West_Point|West Point]] strategist [[w:John_Spencer|John Spencer]] writing in the WSJ to avoid four common foreign-policy fallacies: * the "abacus fallacy" that wars are won by superior resources, counterexample Vietnam * the "vampire fallacy" that wars are won by superior technology, counterexample Russia's failure in Ukraine, (c.f. Lt. Gen [[w:H.R._McMaster|H.R. McMaster]], 2014) * the "Zero Dark Thirty" fallacy that elevates precision strikes and special ops to the level of grand strategy or above (ibid) * and the "Peace table fallacy", which believes that all wars end in negotiation.<ref>{{Cite news|url=https://www.wsj.com/opinion/stopping-endless-wars-is-easier-said-than-done-trump-second-term-2cab9c7a?page=1|title=Stopping ‘Endless Wars’ Is Easier Said Than Done|last=Spencer|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> === Department of State === {{Main article|w:Second presidency of Donald Trump#Prospective foreign policy|w:State Department}} [[w:Marco_Rubio|Marco Rubio]] has been nominated as [[w:U.S._Secretary_of_State|U.S. Secretary of State]], overseeing $53bn and 77,880 employees ==== [[w:USAID|USAID]] ==== ==== National Endowment for Democracy ==== The [[w:National_Endowment_for_Democracy|National Endowment for Democracy]] is grant-making foundation organized as a private non-profit corporation overseen by congress, a project of Ronald Reagan announced in a [[1982 speech to British Parliament]], in which he stated that "freedom is not the sole prerogative of a lucky few, but the inalienable and universal right of all human beings", invoking the Israelites exodus and the Greeks' stand at Thermopylae. The NED is reportedly near the top of the DOGE's hit list.<ref>{{Cite news|url=https://www.wsj.com/opinion/save-a-reagan-initiative-from-the-doge-national-endowment-for-democracy-funding-2b6cc072?page=1|title=Save a Reagan Initiative From Musk and Ramaswamy|last=Galston|first=William|date=11 December 2024|work=The Wall Street Journal}}</ref> It is had a pro-freedom and [[w:Anti-communist|anti-communist]] mission to help pro-democracy leaders and groups in Asia, Africa, and Latin America, and assisted the transition of Eastern and Central European nations. The arguments being made by those in favor of defunding are that it "is a relic of the Cold War that has outlived its usefulness and no longer serves any pressing purpose in terms of advancing national interests", according to [[w:James_Piereson|James Piereson]]{{Cn}} Congress has raised its funding significanty in recent years, in a vote of confidence. === U.S. Department of Defense === U.S. DoD employees ____ civilian personel, ___ civilian contractors, and oversees a budget of _____. Nominated for [[w:Secretary_of_Defense|Secretary of Defense]] is [[w:Pete_Hegseth|Pete Hegseth]], who has been doubted by many Republican Senators{{Cn}} and supported by Trump's base.{{Cn}} The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === [[w:DARPA|DARPA]] === === US Air Force === Air Force is advancing a program called [[w:Collaborative_Combat_Aircraft|Collaborative Combat Aircraft]] to build roughly 1,000 UAVs, with [[w:Anduril|Anduril]] and [[w:General_Atomics|General Atomics]] currently building prototypes, ahead of an Air Force decision on which company or companies will be contracted to build it. The cost quickly exceeded the $2.3 billion approved for last fiscal year’s budget, according to the [[w:Congressional_Research_Service|Congressional Research Service]], prompting calls for more oversight. “''If you want to make real improvements to the defense and security of the United States of America, we would be investing more in drones, we’d be investing more in [[w:Hypersonic_weapon|hypersonic missiles]]'',” said Mr. Ramaswamy. The program for Lockheed-Martin's [[w:F-35|F-35]] stealth jet fighters, now in production, is expected to exceed $2 trillion over several decades. The Air Force on 5 December announced it would delay a decision on which company would build the next-generation crewed fighter, called [[w:Next Generation Air Dominance|Next Generation Air Dominance (NGAD)]], which was planned to replace the [[w:F-22|F-22]] and operate alongside the F-35. Mr. Musk has written that "manned fighter jets are obsolete in the age of drones.” In another post, he claimed “a reusable drone” can do everything a jet fighter can do “without all the overhead of a pilot.” Brigadier General [[w:Doug_Wickert|Doug Wickert]] said in response, “There may be some day when we can completely rely on roboticized warfare but we are a century away.... How long have we thought full self-driving was going to be on the Tesla?” <ref>{{Cite news|url=https://www.wsj.com/politics/national-security/air-force-jets-vs-drones-trump-administration-8b1620a5?page=1|title=Trump Administration Set to Decide Future of Jet Fighters|last=Seligman|first=Lara|date=6 December 2024|work=The Wall Street Journal}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Space Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. === Rail and Tunnel Authority === === Ports Authority === === Interstate Highway Authority === === [[w:Federal_Aviation_Administration|Federal Aviation Administration]] === Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} === National Air and Space Administration === The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] (NASA) had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization." In an interview Isaacman said that NASA will evolve as private space companies set their own priorities and develop technology. NASA could have a certification role for astronauts and vehicles, similar to how the Federal Aviation Administration oversees the commercial airline industry. “The FAA doesn’t build the airplanes. They don’t staff the pilots that fly you from point A to B,” he said. “That is the world that NASA is in, essentially.” He also suggested openness to new and lower cost ways of getting to the Moon and to Mars.<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} === National Oceanic and Atmospheric Administration === == Department of Education and Propaganda == [[w:United_States_Department_of_Education|Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> Nominated as [[w:US_Secretary_of_Education|Secretary of Education]] is [[w:Linda_McMahon|Linda McMahon]]. The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> During his campaign, Trump had pledged to get the "transgender insanity the hell out of schools.” Relying on the district court's decision in ''[[w:Tatel_v._Mount_Lebanon_School_District|Tatel v. Mount Lebanon School District]] , the'' attorney general and education secretary could issue a letter explaining how enforcing gender ideology violates constitutional [[w:Free_exercise_clause|First amendment right to free exercise of religion]] and the [[w:Equal_Protection_Clause|14th Amendment’s Equal Protection Clause]].<ref>{{Cite news|url=https://www.wsj.com/opinion/how-trump-can-target-transgenderism-in-schools-law-policy-education-369537a7?page=1|title=How Trump Can Target Transgenderism in Schools|last=Eden|first=Max|date=9 December 2024|work=The Wall Street Journal}}</ref> === CPB, PBS, NPR === Regarding the [[w:Corporation_for_Public_Broadcasting|Corporation for Public Broadcasting]], [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> {{As of|2024|alt=For fiscal year 2024}}, its appropriation was US$525 million, including $10 million in interest earned. The distribution of these funds was as follows:<ref>{{cite web|url=https://cpb.org/aboutcpb/financials/budget/|title=CPB Operating Budget|last=|date=2024|website=www.cpb.org|archive-url=|archive-date=|access-date=November 27, 2024|url-status=}}</ref> * $262.83M for direct grants to local public television stations; * $95.11M for television programming grants; * $81.77M for direct grants to local public radio stations; * $28.12M for the Radio National Program Production and Acquisition * $9.43M for the Radio Program Fund * $31.50 for system support * $26.25 for administration Public broadcasting stations are funded by a combination of private donations from listeners and viewers, foundations and corporations. Funding for public television comes in roughly equal parts from government (at all levels) and the private sector.<ref>{{cite web|url=http://www.cpb.org/annualreports/2013/|title=CPB 2013 Annual Report|website=www.cpb.org|archive-url=https://web.archive.org/web/20160212170045/http://cpb.org/annualreports/2013/|archive-date=February 12, 2016|access-date=May 4, 2018|url-status=dead}}</ref> == Department of Justice (DOJ) == The US. [[w:Department_of_Justice|Department of Justice]] has a 2023 budget of _____ and ___ employees. Nominated as [[w:Attorney_General|Attorney General]] is Florida AG [[w:Pam_Bondi|Pam Bondi]]{{Cn}}, after Matt Gaetz withdrew his candidacy after pressure.{{Cn}} === Federal Bureau of Investigation (FBI) === With 35,000 employees the [[w:FBI|FBI]] made a 2021 budget request for $9.8 billion. Nominated as [[w:Director_of_the_Federal_Bureau_of_Investigation|FBI Director]] is [[w:Kash_Patel|Kash Patel]], who promised to "shut down the FBI [[w:Hoover_building|Hoover building]] on day one, and open it the next day as a museum of the deep state. He said "''I would take the 7,000 employees that work in that building and send them out across the America to chase criminals"'', saying ''"Go be cops."'' He promised to retaliate against journalists and government employees who "helped Joe Biden rig the election" in 2020.<ref>{{Cite web|url=https://www.wsj.com/video/series/wsj-explains/who-is-kash-patel-donald-trumps-pick-to-lead-the-fbi/F4D38D41-013A-4B05-A170-D7394AA91C2B|title=Who Is Kash Patel, Donald Trump’s Pick to Lead the FBI?|date=6 December 2024|website=WSJ.com Video}}</ref> Director [[w:Christopher_A._Wray|Christopher A. Wray]] announced 11 December that he would step down. [[w:Deputy_Director_of_the_Federal_Bureau_of_Investigation|Deputy]] [[w:Paul_Abbate|Paul Abbate]] will be the interim director, saying “In my view, this is the best way to avoid dragging the Bureau deeper into the fray." Mr. Trump said "the resignation of Christopher Wray is a great day for America".<ref>{{Cite news|url=https://www.wsj.com/politics/policy/christopher-wray-fbi-director-resigns-69069f42?page=1|title=Christopher Wray to Step Down as FBI Director|last=Gurman|first=Sadie|date=11 December 2024|work=The Wall Street Journal}}</ref> ===== Reception and Analysis ===== His nomination "sent shock waves" through the DOJ, and his nomination has been opposed by many Republican lawmakers{{Cn}}, including former CIA director [[w:Gina_Haspell|Gina Haspell]] and AG [[w:Willliam_Barr|Willliam Barr]], who had threatened to resign if Mr. Patel were to be forced on them as a deputy, during Mr. Trump's first term.<ref name=":6">{{Cite news|url=https://www.wsj.com/opinion/kash-patel-doesnt-belong-at-the-fbi-cabinet-nominee-5ef655eb?page=1|title=Kash Patel Doesn’t Belong at the FBI: At the NSC, he was less interested in his assigned duties than in proving his loyalty to Donald Trump.|last=Bolton|first=John|date=11 December 2024|work=The Wall Street Journal}}</ref> As an author he wrote a polemical children's book lionizing "King Donald" with himself in the role of "Wizard", despite the Constitution's republican values and its [[w:Foreign_Emoluments_Clause|emoluments clause]], which prohibits both granting and receiving titles of nobility.<ref>Original synthethic statement, with constitutional context provided by <nowiki>[[User:Jaredscribe]]</nowiki></ref> He has been accused of exaggerating his roles and accomplishments, and deliberate vowing to violate the [[w:Article_Two_of_the_United_States_Constitution#Clause_5:_Caring_for_the_faithful_execution_of_the_law|"Take care" clause of Article II.3]], "''that the Laws be faithfully executed"'' by placing personal loyalties, vendettas, and hunches above his oath to the Constitution.<ref name=":6" /> He has also been accused of lying about national intelligence by [[w:Mark_Esper|Mark Esper]] in his memoir, recently again by Pence aide [[w:Olivia_Troye|Olivia Troye]], although former SoS [[w:Mike_Pompeo|Mike Pompeo]] has not yet clarified the incident in question.  He was called upon to do so in a 11 December WSJ piece by former National Security Advisor [[w:John_Bolton|John Bolton]], who also wrote, "''If illegitimate partisan prosecutions were launched [by the Biden administration], then those responsible should be held accountable in a reasoned, professional manner, not in a counter-witch hunt.  The worst response is for Mr. Trump to engage in the prosecutorial [mis]conduct he condemns [which further] politicizes and degrades the American people's faith in evenhanded law enforcement.''"<ref name=":6" /> He has received support from _____ who wrote that _______.{{Cn}} == U.S. Department of Health and Human Services (DHHS) == Nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United States Secretary of Health and Human Services]] is [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]], deputy Secretary [[w:Jim_O'Neill_(investor)|Jim O'Neill]]. Mr. Kennedy warned on 25 October that the FDA's "war on public health is about to end", accusing it of suppressing psychedelics, stem cells, raw milk, hydroxycloroquine, sunshine, and "anything else that advances human health and can't be patented by Pharma." He said that on day one he would "advise all US water systems to remove fluoride from public water".<ref>{{Cite news|title=MAHA man|date=9 November 2024|work=The Economist|department=News editors}}</ref> Nominated for [[w:US_Surgeon_General|US Surgeon General]] is [[w:Janette_Nesheiwat|Janette Nesheiwat]]. [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Nominee !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Marty_Makary|Marty Makary]] |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- | |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- | |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Dave_Weldon|Dave Weldon]] |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:Jay_Bhattacharya|Jay Bhattacharya]] |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- | |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- | |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Mehmet_Oz|Mehmet Oz]] |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- | |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- | |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === The [[w:NIH|NIH]] distributes grants of ~$50bn per year. Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]], who has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic", due to a phenomenon has been called [[Eroom’s law]], which explains that career incentives encourage “me-too research,” given that citations by other scientists “have become the dominant way to evaluate scientific contributions and scientists.” That has shifted research “toward incremental science and away from exploratory projects that are more likely to fail, but which are the fuel for future breakthroughs.”<ref name=":3">{{Cite news|url=https://www.wsj.com/opinion/jay-bhattacharya-and-the-vindication-of-the-fringe-scientists-pandemic-lockdowns-38b6aec6|title=Jay Bhattacharya and the Vindication of the ‘Fringe’ Scientists|last=Finley|first=Allysia|date=1 December 2024|work=Wall Street Journal}}</ref> Dr. Bhattacharya's February 2020 paper explaining Eroom's law, as possible explanation for slowing of pharmaceutical advances.{{Cn}} * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]], encouraging academic freedom among NIH scientists and term limits for NIH leaders. “Those kinds of reforms, I think every scientist would agree, every American would agree, it’s how you turn the NIH from something that is sort of how to control society, into something that is aimed at the discovery of truth to improve the health of Americans,” he said.<ref name=":4">{{Cite news|url=https://www.wsj.com/health/healthcare/covid-lockdown-critic-jay-bhattacharya-chosen-to-lead-nih-2958e5e2?page=1|title=Covid-Lockdown Critic Jay Bhattacharya Chosen to Lead NIH|last=Whyte|first=Liz Essley|date=26 November 2024|work=The Wall Street Journal}}</ref> * Refocusing on research on [[w:Chronic_diseases|chronic diseases]], which is underfunded, and away from [[w:Infectious_diseases|infectious diseases]], which is overfunded. * Ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague [[w:Eran_Bendavid|Eran Bendavid]], arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}&lt;nowiki&gt;</ref> Dr. Bhattacharya, [[w:Martin_Kulldorff|Martin Kulldorff]], then at Harvard, and Oxford’s [[w:Sunetra_Gupta|Sunetra Gupta]] formally expounded this idea in the [[w:Great_Barrington_Declaration|Great Barrington Declaration]] in October 2020, urging the government to focus on protecting the vulnerable while letting others go about their lives, which previous NIH director [[w:Francis_Collins|Francis Collins]] derided as "fringe science its into the political views of certain parts of our confused political establishment," and previous [[w:NIAID|NIAID]] director [[w:Chief_Medical_Advisor_to_the_President|chief medical advisor to the President]] [[w:Anthony_Fauci|Anthony Fauci]] "a quick and devastating public takedown of its premises." Some suggest the same career incentives that lead to scientific group-think in the pharmaceutical industry, also explain conformist behavior during COVID-19, due to the threat against young scientists of losing NIH funding, jobs, and career opportunities, if they were to exercise in independent judgement.<ref name=":3" /> “Dr. Jay Bhattacharya is the ideal leader to restore NIH as the international template for gold-standard science and evidence-based medicine,” DHHS Secretary nominee Kennedy wrote. "We will reform American scientific institutions so that they are worthy of trust again and will deploy the fruits of excellent science to make America healthy again!” said Dr. Bhattacharya. “Dr. Bhattacharya is a strong choice to lead the NIH,” said Dr. [[w:Ned_Sharpless|Ned Sharpless]], a former [[w:National_Cancer_Institute|National Cancer Institute]] director. “The support of moderate Senate Republicans will be critical to NIH funding, and Dr. Bhattacharya’s Covid work will give him credibility with this constituency.”<ref name=":4" /> === Food and Drug Administration === The FDA in 2022 had 18,000 employees<ref name="fy2022">{{cite web|url=https://www.fda.gov/media/149613/download|title=FY 2022 FDA Budget Request|publisher=FDA|archive-url=https://web.archive.org/web/20230602090805/https://www.fda.gov/media/149613/download|archive-date=June 2, 2023|access-date=January 14, 2022|url-status=live}}</ref> and a budget of $6.5{{nbsp}}billion (2022)<ref name="fy2022" /> Nominated as director is [[w:Marty_Makary|Marty Makary]]. == Department of Agriculture and Food == [[w:Department_of_Agriculture|Department of Agriculture]] (USDA) had 2023 budget of ___ and ____ employees. Nominated for [[w:Secretary_of_Agriculture|Secretary of Agriculture]] is [[w:Brooke_Rollins|Brooke Rollins]], who had earlier served on the [[w:Office_of_American_Innovation|Office of American Innovation]] under [[w:Jared_Kushner|Jared Kushner]], and served as director of [[w:Domestic_Policy_Council|Domestic Policy Council]]. She has not endorsed the "[[w:Make_America_Healthy_Again|Make America Healthy Again]]" agenda of RFK Jr. (and his colleagues Jay Bhattacharcya and others) who promised to "reverse 80 years of farm policy" and complains of the $30 billion/year farms subsidies. Kennedy wants to remove soda from food aid, and ultra-processed food from both [[w:Food-stamps|food-stamp]] benefits and [[w:School_meals|school meals]], both of which are overseen by the USDA; an effort that in the past has been opposed by the food industry, lawmakers, and some anti-hunger advocacy groups.<ref name=":5" /> RFK Jr.'s team had recommended [[w:Sid_Miller|Sid Miller]] for the role, and a group of farmers he had asked to vet candidates had proposed [[w:John_Kempf|John Kempf]].<ref name=":5">{{Cite news|url=https://www.wsj.com/politics/policy/trump-agriculture-pick-brooke-rollins-rfk-jr-1a85beda?page=1|title=RFK Jr. Team Skeptical About USDA Pick|last=Andrews|first=Natalie|date=11 December 2024|work=The Wall Street Journal|others=et al}}</ref> He has also called for re-examining the the [[standards regulating the use of pesticides]], especially [[w:Glyphosate|glyphosate]], the world's most widely used [[w:Herbicide|herbicide]] and the active ingredient in [[w:Roundup|Roundup]], used as a weedkiller in major [[w:U.S._commodity_crops|U.S. commodity crops]].[[w:Herbicide|herbicide]]<ref name=":5" /> === Food Stamps === === School Meals === === Farm Subsidies === == Department of Treasury and Reserve == [[w:Departments_of_Treasury|Departments of Treasury]] has 2023 budget of ____ and ___ employees. Nominated for [[w:Secretary_of_the_Treasury|Secretary of the Treasury]] is [[w:Scott_Bessent|Scott Bessent]]. === Consumer Financial Protection Bureau === The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> === Securities and Exchange Commission === [[w:Securities_and_Exchange_Commission|Securities and Exchange Commission]] === Internal Revenue Service === [[w:Internal_Revenue_Service|Internal Revenue Service]] === Federal Reserve === Mr. Musk has suggested starting a "[[w:Sovereign_Wealth_Fund|Sovereign Wealth Fund]]" like Texas and other U.S. states, instead of hosting a [[w:National_debt|national debt]]. Ron Paul and others have called for abolishing America's [[w:Central_Bank|Central Bank]], the [[w:Federal_Reserve|Federal Reserve System]], which Mr. Musk appeared to endorse. === American Sovereign Wealth Fund === == Department of Industry, Labor, and Commerce == [[w:Department_of_Commerce|Department of Commerce]] has 2023 budget of _____ and _____ employees. Nominated as [[w:Secretary_of_Commerce|Secretary of Commerce]] is [[w:Howard_Lutnick|Howard Lutnick]] [[w:Department_of_Labor|Department of Labor]] has 2023 budget of _____ and ____ employees. Nominated as [[w:Secretary_of_Labor|Secretary of Labor]] is [[w:Lori_Chavez-Remer|Lori Chavez-Remer]] == Departments of Energy and Interior == Nominee for [[w:US_Secretary_of_the_Interior|Secretary of the Interior]] is [[w:Doug_Burgum|Doug Burgum]], who will also be [[w:List_of_U.S._executive_branch_czars|Energy Czar]]. [[w:Department_of_Energy|Department of Energy]] [[w:United_States_Secretary_of_Energy|secretary nominee]] [[w:Chris_Wright_(energy_executive)|Chris Wright]] admits that burning fossil fuels contributes to rising temperatures, but says it poses only a modest threat to humanity, and praises it for increasing plant growth, making the planet greener, and boosting agricultural productivity. He also says that it likely reduces the annual number of temperature-related deaths. (estimates from health researchers say otherwise). He says, "It's probably almost as many positive changes as negative changes... Is it a crisis, is it the world's greatest challenge, or a big threat to the next generation? No. .. A little bit warmer isn’t a threat. If we were 5, 7, 8, 10 degrees [Celsius] warmer, that would be meaningful changes to the planet.” Scientists see a 1.5 degrees Celsius temperature as creating potentially irreversible changes for the planet, and expect to pass that mark later this year, after increasing over several decade. He criticizes the [[w:Paris_climate_agreement|Paris climate agreement]] for empowering "political actors with anti-fossil fuel agendas." Wright favors development of [[w:Geothermal_energy|geothermal energy]] and [[w:Nuclear_energy_policy_of_the_United_States|nuclear energy]], criticizing subsidies to wind and solar energy. <ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/politics/policy/who-is-chris-wright-trump-energy-secretary-9eb617dc?page=1|title=Trump’s Energy Secretary Pick Preaches the Benefits of Climate Change|last=Morenne|first=Benoit|date=9 December 2024|work=The Wall Street Journal}}&lt;nowiki&gt;</ref> === Bureau of Land Management === === Forest Service === === National Parks === == AI and Cryptocurrency Policy == [[w:David_Sacks|David Sacks]] was named "White House AI and Crypto Czar". == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Government Efficiency Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref name=":2"><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}</ref> [[w:Marc_Andreesen|Marc Andreesen]] has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. [[w:Jared_Birchall|Jared Birchall]] has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" [[w:Shaun_MacGuire|Shaun MacGuire]] has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. [[w:Antonio_Gracias|Antonio Gracias]] and [[w:Steve_Davis|Steve Davis]] from Musk's "crisis team" have been active, as has investor [[w:John_Hering|John Hering]]. Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * [[w:Mark_Pincus|Mark Pincus]] * [[w:David_Marcus|David Marcus]] * [[w:Barry_Akis|Barry Akis]] * [[w:Shervin_Pishevar|Shervin Pishevar]], who has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions."<ref name=":2" /> [[w:William_McGinley|William McGinley]] will move to a role with DOGE. Originally nominated for [[w:White_House_counsel|White House counsel]], he will be replaced in that role by [[w:David_Warrington|David Warrington]].<ref>{{Cite news|title=A White House Counsel Replaced before starting|last=Haberman|first=Maggie|date=6 December 2024|work=New York times}}</ref> The WSJ lauded without naming them, comparing them to the "dollar-a-year men" - business leaders who during WWII revolutionized industrial production to help make America the "arsenal of democracy". (WSJ, 10 December 2024) == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[w:Ministry_of_Deregulation_and_State_Transformation|Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] * [[w:Delivering_Outstanding_Government_Efficiency_Caucus|Delivering Outstanding Government Efficiency Caucus]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} htx071rbn2rpz0vdsfyo0wnwelw2abd 0 316554 2691665 2688748 2024-12-12T17:13:28Z Dan Polansky 33469 2691665 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at what is called generative artificial intelligence (GenAI) and large language models (LLM). Examples include ChatGPT, Gemini, Copilot and LLaMA. There are benefits, there are risks and there are costs. An immediately obvious risk is inaccuracy. GenAI can easily generate inaccurate/untrue statements. This can be mitigated by awareness of the user. After all, users need to learn critical attitute toward sources that they read anyway; GenAI is far from the only offender as for being source of untrue statements. A benefit is the use of GenAI as a source of ideas to be indepedently examined or verified. One use of this is for initial statement verification/probing: one can e.g. ask 'Is the following accurate: "Adjectives are never capitalized in English."' and have the statement corrected. However, the above mentioned risk really needs an emphasis. It seems all too easy and tempting to trust the answer without independent verification. One may wonder whether GenAI can be used as a form of psychotherapy. A remarkable feature is the limitless patience shown in answering questions, even stupid or annoying questions. One can practice asking questions, improving formulations of questions, thinking critically about the answers, etc. GenAI can be charged to contribute to global climatic change via electricity use. The ethics of this aspect is for each prospective user to consider; governments have not prohibited GenAI for this reason and seem unlikely to do so, provided they did not for the most part even prohibitd cryptocurrency/cryptoasset mining. A serious analysis of this aspect would include a quantitative comparison of other dispensable uses of energy such as video streaming. GenAI can also draw/paint/create images based on verbal description. For this use, the label large language model seems misleading or inaccurate, on the face of it. Interestingly, GenAI seems rather inapt in even trivial calculation, as per Edmund Weitz video. Tools providing complementary facilities to GenAI are e.g. Wolfram Alpha and Desmos Calculator. It would be interesting to see what would happen if one could somehow integrate genAI with e.g. Wolfram Alpha, that is, when genAI would delegate computational assignments to Wolfram Alpha (or equivalent). One can ask whether the label generative artificial intelligence is appropriate. That is, one can ask whether this really is an intelligence, one that is artificial and generative. Very superficially, something suggestive of human verbal intelligence is there. Moreover, given the term artificial general intelligence (AGI), we may use the term artificial intelligence much more broadly to include specialized problem/task solving, and then, chess playing would be artificial intelligence. Generative artificial intelligence may even approach passing the Turing test. Paradoxically, the responses from GenAI are too fast to be human, which betrays the artificial origin. Be it as it may, GenAI does not really seem to understand what it is saying; but then, as a sinister note, too many humans speak as if they did not understand what they are saying either. And then, one may wonder whether part of the human brain does not really implement something like GenAI (such an idea is found e.g. [https://www.linkedin.com/pulse/why-you-more-like-genai-than-think-less-so-geoffrey-moore-gdxnc here]). As for the mechanism of function, sources seem to indicate that textual GenAI just tries to determine the next word given the sequence of words (using artificial neural networks). I struggle to find this plausible and to understand how that principle could possibly produce the kind of behavior that we see, but what do I know. I would find it much more plausible if somewhether in the guts of textual GenAI, there would be something like OpenCyc ontology. As for plagiarism. It seems to me that GenAI generally commits plagiarism: it does not attribute the sources from which it takes ideas. Plagiarism is not the same concept as copyright violation. Plagiarism is the use of ideas obtained from sources without attribution and presenting them as one's own. Some form of plagiarism is perhaps widespread anyway; attributing all ideas to sources seem to be a rather stringent requirement. One defense could be this: GenAI does not represent to have any ideas of its own; it attributes all ideas to sources, albeit unspecified ones. But from what I understand, failing to specify sources from which ideas are taken is still plagiarism. See also [[One man's look at copyright law]]. == See also == * [[Should Wikiversity allow editors to post content generated by LLMs?]] * [[Wikiversity:Artificial intelligence]] * [[Motivation and emotion/Assessment/Using generative AI]] == Further reading == * {{W|Generative artificial intelligence}}, wikipedia.org * [https://www.youtube.com/watch?v=medmEMktMlQ ChatGPT und die Mathematik] by Edmund Weitz, youtube.com (in German) * [https://www.youtube.com/watch?v=5cYYeuwYF_0 ChatGPT und die Logik] by Edmund Weitz, youtube.com (in German) [[Category:Generative artificial intelligence]] 4woz8jrk3r4kgwqhz4g1uju8egpksqc 0 316762 2691653 2690451 2024-12-12T16:39:30Z DavidMCEddy 218607 /* Declaring war on bathtubs */ add reference re bathtubs 2691653 wikitext text/x-wiki :''This is a discussion of an interview 2024-11-21 with Simon Fraser University professor Ahmed Al-Rawi<ref name=AlRawiSFU><!--SFU homepage of Ahmed Al-Rawi-->{{cite Q|Q131349551}}</ref> about his research into how to understand and counter the rise in political polarization and violence worldwide. A 29:00 mm:ss podcast excerpted from the companion video will be posted here after it is released to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref>'' [[File:Media literacy to counter political polarization per Ahmed Al-Rawi.webm|thumb|Media literacy to counter political polarization: Interview with [[w:Simon Fraser University|Simon Fraser University]] professor Ahmed Al-Rawi about media and conflict.]] <!--[[File: ... .ogg|thumb|29:00 mm:ss extract from interview recorded 2024-10-25 regarding the legal concerns of Wikimedia Europe.]]--> [[w:Simon Fraser University|Simon Fraser University]] professor Ahmed Al-Rawi<ref name=AlRawiSFU/> discusses the media literacy laboratory he co-founded at the Lebanese American University in Beiruit<ref><!-- Author archives: Ahmed Al-Rawi, LLRX-->{{cite Q|Q131349668}}</ref> and his research into how to understand and counter the rise in political polarization and violence worldwide. He is interviewed by Spencer Graves. Al-Rawi is the author or co-author of a dozen books in the last dozen years plus co-editor of three others and author of dozens of articles.<ref name=AlRawiCV><!--Curriculum Vitae: Ahmed Al-Rawi-->{{cite Q|Q131349693}}</ref> Most of his publications describe the increase in political polarization and violence worldwide in recent decades and what might be done to counter it. His research has focused primarily on the Arab World and on Canada. At Simon Fraser and elsewhere he has taught classes on media, communications, democracy and power. Al-Rawi is currently an Associate Professor of News, Social Media & Public Communication in the School of Communication, Faculty of Communication, Art & Technology at Simon Fraser University in [[w:Vancouver|Vancouver]], British Columbia, Canada and a scientist with the [[w:International Panel on the Information Environment|International Panel on the Information Environment]]<ref><!--Ahmed Al-Rawi, IPIE-->{{cite Q|Q131349735}}</ref> He has previously taught at other universities in Canada as well as in the [[w:Netherlands|Netherlands]] and in [[w:Oman|Oman]]. Twenty years ago he worked as a freelance radio journalist for the Pacifica Radio Network and before that as a translator for Iraq National Television, [[w:Baghdad|Baghdad]], Iraq. == Cognitive dissonance == Al-Rawi mentioned, "[[w:cognitive dissonance|cognitive dissonance]], which means people tend to avoid any kind of information that would cause them headache, cause them disruption. They would tend to seek information from sources that would align with their own ways of thinking, with their own values.<ref>Starting around 11:27 mm:ss [[:File:Media literacy to counter political polarization per Ahmed Al-Rawi.webm|in the companion video.]]</ref> That's a theory, a well-known theory and social psychology. So it makes a lot of sense that we see this kind of polarization happening in the media sphere as well." == External pluralism == He also discussed [[w:Media pluralism|media theory called "external pluralism]], where you have separate media outlets that are distinctively different from each other." The US has ''[[w:Breitbart News|Breitbart]]'', ''[[w:Fox News|Fox News]]'', ''[[w:American one|America One]]'' Network, and other networks that are deemed to be more liberal. This is also happening in Canada, Europe and everywhere in the Middle East. Iraq has a very pluralistic media scene situated along the political positions of political and religious parties. Al-Rawi insists this is why "we need more religious and political literacy that could be embedded into media literacy, education, so that people will become more aware about the goals of politicians, messages: What is behind that message?" Greater public awareness of these issues could avoid escalation of tensions. Graves asked about the role of the media [[w:Yemen civil war|in the war in Yemen]]. Al-Rawi noted that the different parties to that conflict including Saudi Arabia and Iran each have their own media outlets promoting their own agendas exacerbating the conflict and eliminating possible solutions. Graves asked about the [[Winning the War on Terror|role of the media in the war on terror]]. Al-Rawi had earlier mentioned the role of the major media in the US in parroting what the administration was saying. Graves suggested that any responsible journalist should have known at the time that the official justification for the invasion of Iraq in 2003 was at best questionable and likely fraudulent, as it turned out to be. Al-Rawi replied that this was one of the most "significant turning points in the lives of Muslim Americans, even Muslim Canadians" and Muslims living in Europe, with many forced to leave even though they had done nothing wrong. == Declaring war on bathtubs == Graves noted that Obama as president commented that more Americans have died in the average year [except for 2001] drowning in bathtubs, hot tubs and spas than have succumbed to terrorism.<ref>As noted in the section on [[Winning the War on Terror#2.11. US foreign interventions in opposition to democracy|"US foreign interventions in opposition to democracy"]] in the Wikiversity article on "[[Winning the War on Terror]]", 'On May 23, 2013, then-US President Obama noted that terrorism caused fewer American deaths than car accidents or falls in the bathtub. He occasionally ''had to be badgered by advisors into choices commensurate with popular fear.'' He worried, too, that counterterrorist priorities “swamped” his other foreign policy aspirations.' (Per <!-- Humane: How the United States Abandoned Peace and Reinvented War -->{{cite Q|Q108896140}}, pp. 268, 299-300.)</ref> Al-Rawi agreed: "A lot of people lost their lives, and it's not their fault." In "the war that happened after 2003, the number of people dying every month is like we have a 9-11 every 2 or 3 weeks. ... These were mostly innocent civilians. ... We have the same thing happening, for example, in Gaza and elsewhere." == The relative effectiveness of law enforcement vs. military in combatting terrorism == Graves noted that, "Jones and Libicki found 268 terrorist groups that ended between 1968 and 2,006, 43% ended with negotiations like the Good Friday agreement in Northern Ireland, 40% were taken out by law enforcement. 10% won. 3% were defeated militarily. Yet the United States is promoting the absolute, least effective approach to terrorism."<ref>Jones and Libicki (2008).</ref> Al-Rawi agreed: "Force is not the answer to ending terrorism. Terrorism is an idea. If you want to kill the idea, you have to use another idea." == Talking politics == Al-Rawi noted that, "if you are talking to someone who is trying to enhance political polarization, it's really important to understand where their concerns come from, whether these are imagined or real, because that's where you can actually find a common ground or an alternative identity, whereby you can have a mutual discussion and probably reach some kind of consensus harmony, ... that would at least lessen this kind of polarization." == The threat == Internet company executives have knowingly increased political polarization and violence including the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise might have reduced their profits. Documentation of this is summarized in [[:Category:Media reform to improve democracy]]. == Media effect theory == Al-Rawi also discussed "[[w:Influence of mass media|media effect theory]]", which describes how the media influence the thoughts, attitudes, and behaviors of individuals in their audience. It sets the agenda for what many people think about. This effect could be both negative, increasing political polarization and the risk of violence, but also positive by focussing on shared identities.<ref>Starting around 15:47 mm:ss in [[:File:Media literacy to counter political polarization per Ahmed Al-Rawi.webm|the accompanying video]].</ref> ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Ahmed Al-Rawi (2025, forthcoming) Mediated Racism and democracy in Canada: Interrogating the news industry, political systems, and public discourses, Routledge-->{{cite Q|Q131349901|date=2025b}} * <!-- Ahmed Al-Rawi (2025, forthcoming) Disruptive Information in Canada, Bloomsbury-->{{cite Q|Q131349920|date=2025a}} * <!-- Ahmed Al-Rawi et al. (2025) The Canadian Far-Right and Conspiracy Theories, Routledge-->{{cite Q|Q131349937}} * <!-- Ahmed Al-Rawi (2024) The Iraqi Spring: Social Media and Political Activism, Amsterdam U. Pr.-->{{cite Q|Q131350073}} * <!-- Ahmed Al-Rawi (2024) Online hate on social media, Palgrave Macmillan-->{{cite Q|Q131350104}} * <!-- Ahmed Al-Rawi (2024) ISIS' propaganda machine : global mediated terrorism, Routledge-->{{cite Q|Q131350154}} * <!-- Ahmed Al-Rawi (2023) Supernatural Creatures in Arabic Literary Tradition, Routledge-->{{cite Q|Q131350208}} * <!-- Ahmed Al-Rawi (2021) Cyberwars in the Middle East, Routledge-->{{cite Q|Q131350317}} * <!-- Ahmed Al-Rawi (2020) News 2.0: Journalists, Audiences and News on Social Media, Wiley-Blackwell-->{{cite Q|Q131350446}} * <!-- Ahmed Al-Rawi (2020) Women's Activism and New Media in the Arab World, SUNY Pr.-->{{cite Q|Q131350555}} * <!-- Ahmed Al-Rawi (2017) Islam on YouTube : Online Debates, Protests, and Extremism, Palgrave Macmillan-->{{cite Q|Q131350577}} * <!-- Ahmed Al-Rawi (2012) Media Practice in Iraq, Springer-->{{cite Q|Q131350656}} * {{cite Q|Q57515305}}<!-- Jones and Libicki (2008) How terrorist groups end--> [[Category:Politics]] [[Category:Freedom and abundance]] [[Category:Media reform to improve democracy]] 83xogilznf0qc1yyrckekqke5brsaw7 0 316907 2691630 2691514 2024-12-12T13:56:41Z Eshaa2024 2993595 2691630 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] ''curvus'' for "bent", "curved") is a [[w:en:Dimension (mathematics)|one dimensionals]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===Example 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Trace ( \gamma ) := \left\{ \gamma(t) \ | \ a \leq t \leq b \right\} </math>. === Difference - Graph und Curve === For a curve <math>\gamma:[a,b] \to \mathbb{R}^2</math> the Supr or curve is a subset of <math> \mathbb{R}^2 </math>, while the graph of function <math> Graph(\gamma) \subset \mathbb{R}^3 </math> is. === Task - Plot Graph und Curve === use [[CAS4Wiki]] : :<math> \begin{array}{rrcl} \gamma: & [0,6\pi] & \rightarrow & \mathbb{R}^2 \\ & t & \mapsto & \gamma(t) = \left( 3\cdot \cos(t), \sin(t) \right) \end{array} </math> ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. ==== Geogebra - Interactive Implementation ==== Create a value slider in Geogebra with the variable name <math>t</math> and create the following 3 points step by step in the command line of Geogebra and move the value slider for <math>t</math> after that. <code><pre> K_1:(2*cos(t),2 * sin(t)) K_2:(cos(3*t),sin(3*t)) K: K_1+K_2 </pre></code> The construction about will create an interactive representation of the the follow path <math display="inline"> \gamma_4 </math>. Observe the point <math>K</math> in Geogebra. :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Example in Geogebra] ==Representations of Image Sets by Equations == A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:Polynomial|Polynomial]], the curve is called ''[[w:en:Algebraic|algebraic]]''. == Graph of a function== [[w:en:Functiongraph|Functiongraphs]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the[[w:en:Mathematics education|Mathematics education]] of [[w:en:Curve sketching|Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the[[w:en:number of revolutions|number of revolutions]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:circulation theorem|circulation theorem]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. == Curves as Independent Objects == Curves without an ambient space are relatively uninteresting in [[w:en:Differential Geometry]] because every one-dimensional [[w:en:Manifold|manifold]] is [[w:en:Diffeomorphism|diffeomorphic]] to the real line <math> \mathbb{R} </math> or to the unit circle <math>S^1</math>. Also, properties like the [[w:en:Curvature|curvature]] of a curve are intrinsically undetectable. In [[w:en:Algebraic Geometry|algebraic geometry]] and, correspondingly, in [[w:en:Complex Analysis|complex analysis]], "curves" typically refer to one-dimensional [[w:en:Complex Manifold|complex manifolds]], often also called [[w:en:Riemann Surface|Riemann surfaces]]. These curves are independent objects of study, with the most prominent example being [[w:en:Elliptic Curve|elliptic curves]]. See [[w:en:Curve (algebraic geometry)|curve (algebraic geometry)]] == Historical == The first book of [[w:en:Elements (Euclid)|Elements]] by [[w:en:Euclid|Euclid]] began with the definition: :: "A point is that which has no parts. A curve is a length without breadth." This definition can no longer be upheld today because, for example, there are [[w:en:Peano Curve|Peano curves]], i.e., continuous [[w:en:Surjective|surjective]] mappings <math> f\colon\mathbb{R} \to \mathbb{R}^2 </math> that fill the entire plane <math> \mathbb{R}^2 </math>. On the other hand, the [[w:en:Sard's Lemma|Sard's Lemma]] implies that every differentiable curve has zero area, i.e., as Euclid demanded, it truly has ''no breadth.'' == Interactive Representations of Curves in GeoGebra == *[https://www.geogebra.org/m/e3hhdrvq Tangent vector of a curve] in <math>\mathbb{R}^2</math> for a curve <math>\gamma:[a,b]\to \mathbb{R}^2</math> with tangent vector <math>\gamma':[a,b]\to \mathbb{R}^2</math> *[https://www.geogebra.org/m/srmgcsZX Rolling curves Bicycle reflectors] as an example of curves - [[w:en:Cycloid|Cycloid]] *[https://www.geogebra.org/m/ppuvs3ge Rolling curve for Example 2] == See also == *[[w:en:Curve_(mathematics)#Space_curves|Space curves]] in <math>\mathbb{R}^3</math> *[[w:en:Category:Curve_(geometry)|Curves in geometry]] *[[w:en:Curve_(mathematics)#Differentiable_curves,_curvature|Differentiable curves, curvature]] *[[w:en:Cycloid|Cycloid]] == Literature == *Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry''. Birkhäuser, Boston 1997. *Wilhelm Klingenberg: ''A Course in Differential Geometry''. Springer, New York 1978. == References == <references /> == External Links == {{Commonscat|Curves|Kurven}} {{Wiktionary|Kurve}} == Page Information == You can display this learning resource as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie Course:Complex Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. <!-- * The content of the page is based on the following material: ** [https://de.wikipedia.org/wiki/Kurs:Funktionentheorie/Kurven https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven] --> [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven The page] 69gf9zwz81xst7018a2pjx287e0dah2 2691631 2691630 2024-12-12T13:57:21Z Eshaa2024 2993595 /* Literature */ 2691631 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] ''curvus'' for "bent", "curved") is a [[w:en:Dimension (mathematics)|one dimensionals]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===Example 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Trace ( \gamma ) := \left\{ \gamma(t) \ | \ a \leq t \leq b \right\} </math>. === Difference - Graph und Curve === For a curve <math>\gamma:[a,b] \to \mathbb{R}^2</math> the Supr or curve is a subset of <math> \mathbb{R}^2 </math>, while the graph of function <math> Graph(\gamma) \subset \mathbb{R}^3 </math> is. === Task - Plot Graph und Curve === use [[CAS4Wiki]] : :<math> \begin{array}{rrcl} \gamma: & [0,6\pi] & \rightarrow & \mathbb{R}^2 \\ & t & \mapsto & \gamma(t) = \left( 3\cdot \cos(t), \sin(t) \right) \end{array} </math> ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. ==== Geogebra - Interactive Implementation ==== Create a value slider in Geogebra with the variable name <math>t</math> and create the following 3 points step by step in the command line of Geogebra and move the value slider for <math>t</math> after that. <code><pre> K_1:(2*cos(t),2 * sin(t)) K_2:(cos(3*t),sin(3*t)) K: K_1+K_2 </pre></code> The construction about will create an interactive representation of the the follow path <math display="inline"> \gamma_4 </math>. Observe the point <math>K</math> in Geogebra. :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Example in Geogebra] ==Representations of Image Sets by Equations == A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:Polynomial|Polynomial]], the curve is called ''[[w:en:Algebraic|algebraic]]''. == Graph of a function== [[w:en:Functiongraph|Functiongraphs]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the[[w:en:Mathematics education|Mathematics education]] of [[w:en:Curve sketching|Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the[[w:en:number of revolutions|number of revolutions]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:circulation theorem|circulation theorem]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. == Curves as Independent Objects == Curves without an ambient space are relatively uninteresting in [[w:en:Differential Geometry]] because every one-dimensional [[w:en:Manifold|manifold]] is [[w:en:Diffeomorphism|diffeomorphic]] to the real line <math> \mathbb{R} </math> or to the unit circle <math>S^1</math>. Also, properties like the [[w:en:Curvature|curvature]] of a curve are intrinsically undetectable. In [[w:en:Algebraic Geometry|algebraic geometry]] and, correspondingly, in [[w:en:Complex Analysis|complex analysis]], "curves" typically refer to one-dimensional [[w:en:Complex Manifold|complex manifolds]], often also called [[w:en:Riemann Surface|Riemann surfaces]]. These curves are independent objects of study, with the most prominent example being [[w:en:Elliptic Curve|elliptic curves]]. See [[w:en:Curve (algebraic geometry)|curve (algebraic geometry)]] == Historical == The first book of [[w:en:Elements (Euclid)|Elements]] by [[w:en:Euclid|Euclid]] began with the definition: :: "A point is that which has no parts. A curve is a length without breadth." This definition can no longer be upheld today because, for example, there are [[w:en:Peano Curve|Peano curves]], i.e., continuous [[w:en:Surjective|surjective]] mappings <math> f\colon\mathbb{R} \to \mathbb{R}^2 </math> that fill the entire plane <math> \mathbb{R}^2 </math>. On the other hand, the [[w:en:Sard's Lemma|Sard's Lemma]] implies that every differentiable curve has zero area, i.e., as Euclid demanded, it truly has ''no breadth.'' == Interactive Representations of Curves in GeoGebra == *[https://www.geogebra.org/m/e3hhdrvq Tangent vector of a curve] in <math>\mathbb{R}^2</math> for a curve <math>\gamma:[a,b]\to \mathbb{R}^2</math> with tangent vector <math>\gamma':[a,b]\to \mathbb{R}^2</math> *[https://www.geogebra.org/m/srmgcsZX Rolling curves Bicycle reflectors] as an example of curves - [[w:en:Cycloid|Cycloid]] *[https://www.geogebra.org/m/ppuvs3ge Rolling curve for Example 2] == See also == *[[w:en:Curve_(mathematics)#Space_curves|Space curves]] in <math>\mathbb{R}^3</math> *[[w:en:Category:Curve_(geometry)|Curves in geometry]] *[[w:en:Curve_(mathematics)#Differentiable_curves,_curvature|Differentiable curves, curvature]] *[[w:en:Cycloid|Cycloid]] == Literature == *Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry''. Birkhäuser, Boston 1997. *Wilhelm Klingenberg: ''A Course in Differential Geometry''. Springer, New York 1978. == References == <references /> == External Links == {{Commonscat|Curves|Kurven}} {{Wiktionary|Kurve}} == Page Information == You can display this learning resource as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie Course:Complex Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. <!-- * The content of the page is based on the following material: ** [https://de.wikipedia.org/wiki/Kurs:Funktionentheorie/Kurven https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven] --> [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven The page] tix7sxut6rqyera0p07jqiq1de20p2r 0 317026 2691666 2691232 2024-12-12T17:18:34Z Juliafono 2948061 a litlle edition 2691666 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <WikiJournal of Medicine> | last1 = Volpe | orcid1 = 0000-0003-0526-5625 | first1 = Maria Julia Gobbi | last2 = Araujo | orcid2 = 0000-0002-3675-4651 | first2 = Eliene Silva | last3 = Cardoso | orcid3 = 0000-0003-0526-5625 | affiliation2 = Federal University of Rio Grande do Norte, Natal, Brazil | first3 = Maria Julia Ferreira | last4 = Montilha | orcid4 = 0000-0003-2500-4262 | affiliation4 = University of São Paulo, Bauru, Brazil | first5 = Hector Gabriel Corrale | last5 = Matos | orcid5 = 0000-0002-2649-370X | affiliation5 = University of São Paulo, Bauru, Brazil | first6 = Katia de Freitas | last6 = Alvarenga | orcid6 = 0000-0002-7847-3225 | affiliation6 = University of São Paulo, Bauru, Brazil | first7 = Lilian Cássia Bórnia | affiliation3 = University of São Paulo, Bauru, Brazil | first4 = Alexandre Alberto Pascotto | last7 = Jacob | orcid7 = 0000-0003-1947-7506 | affiliation7 = University of São Paulo, Bauru, Brazil | et_al = <!-- if there are >9 authors, hyperlink to the list here --> | affiliation1 = Univertiy of São Paulo, Bauru, Brazil | correspondence1 = mariajuliavolpe@usp.br | correspondence = juliafono2011@gmail.com | keywords = Wiki; Health; Teaching Methodology; university. | license = <CC BY 4.0> | abstract = [[W:Wikipedia | Wikipedia]] is a free, collaborative, and multilingual encyclopedia and corresponds to the largest and most popular reference work on the [[w:Internet|Internet]]. It admits involvement with [[en:w:university education|university education]], enabling the insertion of the student in the active [[en:w:learning |learning methodology]], shuch as allowing [[en:w:Science communication|dissemination of scientific]] content to the population. The aim was to analyze the [[en:w:scientific journals|scientific production]] between 2015 and 2020 on Wikipedia as an [[en:w:active methodology|methodology]] tool for education in [[w:health|health]] and [[w:science communication |scientific dissemination]]. Searches were conducted in November 2020 in the [[w:PubMed |PubMed]], [[w:Latin_American_and_Caribbean_Health_Sciences_Literature |LILACS]], [[w:SciELO |SciELO]], and Portal de Periódicos CAPES databases regarding the use of Wikipedia as an active methodology in health and a tool for scientific dissemination, through crossings between the terms “Wikipedia”, “university” and “health”. The findings associate strategy with the development of scientific research skills, health education, and academic, personal, and collaborative stimuli. Furthermore, Wikipedia refers to high-impact periodicals, and their use contributes to dissemination of current and high-level information to the population. It was possible to identify answers to the guiding questions, favoring the development of academic, professional, and personal skills by Wikipedia as an active methodology of teaching in health, further highlighting its scope as a source of health information. }} ==Introduction== Launched on January 15, 2001, [[w:Wikipedia|Wikipedia]] is a free, collaborative, and multilingual encyclopedia managed by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. It advocates for global empowerment and participation in creating educational content available online under [[w:Creative Commons|open commons licenses]] and in the [[w:Public domain| public domain]], with the goal of promoting widespread sharing<ref name=":0">{{Cite journal|date=2024-10-30|title=Wikipédia|url=https://pt.wikipedia.org/wiki/Wikipedia|journal=Wikipédia, a enciclopédia livre|language=pt}}</ref>. Given that the platform is the largest and most popular general reference work on the internet<ref name=":0" />, it presents the possibility of being adopted in university education. By using Wikipedia to create content grounded in [[w:Scientific evidence|scientific evidence]], students engage in an active and participatory teaching methodology, positioning themselves as active agents in their own [[w:learning|learning process]]<ref>{{Cite journal|last=Mitre|first=Sandra Minardi|last2=Siqueira-Batista|first2=Rodrigo|last3=Girardi-de-Mendonça|first3=José Márcio|last4=Morais-Pinto|first4=Neila Maria de|last5=Meirelles|first5=Cynthia de Almeida Brandão|last6=Pinto-Porto|first6=Cláudia|last7=Moreira|first7=Tânia|last8=Hoffmann|first8=Leandro Marcial Amaral|date=2008-12|title=Metodologias ativas de ensino-aprendizagem na formação profissional em saúde: debates atuais|url=http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1413-81232008000900018&lng=pt&tlng=pt|journal=Ciência & Saúde Coletiva|volume=13|issue=suppl 2|pages=2133–2144|doi=10.1590/S1413-81232008000900018|issn=1413-8123}}</ref><ref>{{Cite journal|date=2007|title=Participatory Action Research (PAR)|url=https://doi.org/10.4135/97814129862681.n251|journal=The SAGE Dictionary of Qualitative Inquiry|location=2455 Teller Road, Thousand Oaks California 91320 United States of America|publisher=SAGE Publications, Inc.|isbn=978-1-4129-0927-3}}</ref><ref>{{Cite book|url=https://doi.org/10.4135/9781473921290.n45|title=Critical Theory and Critical Participatory Action Research|last=Kemmis|first=Stephen|last2=McTaggart|first2=Robin|last3=Nixon|first3=Rhonda|publisher=SAGE Publications Ltd|location=1 Oliver’s Yard, 55 City Road London EC1Y 1SP|pages=453–464}}</ref>. They feel empowered with certain knowledge, allowing them to develop a sense of identity and pass it on to others<ref>WIKI EDUCATION. '''Does editing Wikipedia change a student´s life? WIKI EDU'''. San Francisco, CA: Wiki Education, [2018]</ref>. The active approach involves greater engagement, dedication, and, consequently, responsibility from students in the process of knowledge construction<ref>{{Cite journal|last=Sebold|first=Luciara Fabiane|last2=Martins|first2=Fernanda Espíndola|last3=Da Rosa|first3=Rosiane|last4=Carraro|first4=Telma Elisa|last5=Martini|first5=Jussara Gue|last6=Kempfer|first6=Silvana Silveira|date=2010-12-24|title=METODOLOGIAS ATIVAS: UMA INOVAÇÃO NA DISCIPLINA DE FUNDAMENTOS PARA O CUIDADO PROFISSIONAL DE ENFERMAGEM|url=http://revistas.ufpr.br/cogitare/article/view/20381|journal=Cogitare Enfermagem|volume=15|issue=4|doi=10.5380/ce.v15i4.20381|issn=2176-9133}}</ref>. It also encourages collective work through access to an expanded discussion environment. This leads to improved comprehension of the explored content, as well as the development of [[en:w:communication skills|communication skills]], which are essential for [[en:w:Health professional|healthcare professionals]]<ref>{{Cite journal|last=Amorim|first=Juleimar Soares Coelho de|last2=Poltronieri|first2=Bruno Costa|last3=Ribeiro|first3=Aline Moreira|last4=Ferla|first4=Alcindo Antônio|date=2019|title=Team-based learning in Physical therapy undergraduate course: experiment report|url=https://doi.org/10.1590/1980-5918.032.ao46|journal=Fisioterapia em Movimento|volume=32|doi=10.1590/1980-5918.032.ao46|issn=1980-5918}}</ref>. Furthermore, the platform is widely consulted, with its information accessed by millions of users, particularly in the fields of science and health, by both the general public and professionals in these areas<ref name=":1">{{Cite journal|last=Heilman|first=James M|last2=Kemmann|first2=Eckhard|last3=Bonert|first3=Michael|last4=Chatterjee|first4=Anwesh|last5=Ragar|first5=Brent|last6=Beards|first6=Graham M|last7=Iberri|first7=David J|last8=Harvey|first8=Matthew|last9=Thomas|first9=Brendan|date=2011-01-31|title=Wikipedia: A Key Tool for Global Public Health Promotion|url=http://www.jmir.org/2011/1/e14/|journal=Journal of Medical Internet Research|language=en|volume=13|issue=1|pages=e14|doi=10.2196/jmir.1589|issn=1438-8871}}</ref>. As health-related topics receive significant and growing attention in online searches, these resources can be leveraged to reach the public<ref>{{Cite journal|last=Trotter|first=Matthew I.|last2=Morgan|first2=David W.|date=2008-09|title=Patients' use of the Internet for health related matters: a study of Internet usage in 2000 and 2006|url=https://journals.sagepub.com/doi/10.1177/1081180X08092828|journal=Health Informatics Journal|language=en|volume=14|issue=3|pages=175–181|doi=10.1177/1081180X08092828|issn=1460-4582}}</ref>. In this context, the [[en:w:democratization|democratization]] of scientific knowledge comes into focus, allowing it to reach diverse segments of the population and empowering them, as information becomes accessible for societal use<ref>BIZZOCCHI, A. L. Culture and pleasure: The place of science. '''Cienc Cult''', São Paulo, v. 51, n. 1, p. 26-33, jan/fev 1999. </ref>. [[en:w:Scientific literacy|Scientific literacy]] is recognized as a tool for improving quality of life<ref>{{Cite journal|last=Ribeiro|first=Maria das Graças|last2=Teles|first2=Maria Eloiza de Oliveira|last3=Maruch|first3=Sandra Maria das Graças|date=1995|title=Morphological aspects of the ovary of Columba livia (Gmelin) (Columbidae, Columbiformes)|url=https://doi.org/10.1590/s0101-81751995000100016|journal=Revista Brasileira de Zoologia|volume=12|issue=1|pages=151–157|doi=10.1590/s0101-81751995000100016|issn=0101-8175}}</ref>. Consequently, scientific dissemination is increasingly viewed as both a tool and a [[en:w:social movement|social movement]], fostering [[en:w:citizenship|citizenship]] and enhancing the health of diverse groups<ref>{{Cite journal|last=Bizzo|first=Maria Letícia Galluzzi|date=2002-02|title=Difusão científica, comunicação e saúde|url=https://doi.org/10.1590/s0102-311x2002000100031|journal=Cadernos de Saúde Pública|volume=18|issue=1|pages=307–314|doi=10.1590/s0102-311x2002000100031|issn=0102-311X}}</ref>. Thus, the overarching objective was to analyze scientific production between 2015 and 2020 regarding Wikipedia as a tool for active teaching methodologies in health and scientific dissemination. ==Materials and Methods== A [[en:w:scoping review|scoping review]] was conducted in September 2020. This type of [[en:w:literature review|literature synthesis]] is designed to analyze different types of studies and summarize the available evidence on a topic of interest. The review was structured into the following steps: (i) formulation of the guiding question and research objective; (ii) identification of studies based on the adopted research methodology; (iii) selection of studies using inclusion criteria; (iv) analysis and grouping of data; (v) synthesis of results through a qualitative evaluation of the themes in the articles, based on the research objective and question; and (vi) presentation of results through thematic analysis. The literature review was conducted using the databases available on the CAPES Journals Portal ([[en:w:Coordination for the Improvement of Higher Education Personnel|Coordination for the Improvement of Higher Education Personnel]]), [[en:w:PubMed|PubMed]]/[[en:w:MEDLINE|MEDLINE]], [[en:w:Latin American and Caribbean Health Sciences Literature|LILACS (Latin American and Caribbean Health Sciences Literature)]], and [[en:w:SciELO|SciELO]]. The protocol for this review was registered on the [[en:w:Open Science Framework|Open Science Framework]]<ref>{{Cite journal|last=Cardoso|first=Maria Julia Ferreira|last2=Matos|first2=Hector Gabriel Corrale de|date=2023-09-12|title=Use of Wikipedia at university as a resource of active health teaching methodology and scientific dissemination: scoping review|url=https://osf.io/dwrsj/|language=en|doi=10.17605/OSF.IO/DWRSJ}}</ref> based on the proposal ''How to write a scoping review protocol: Guidance and template''. Any methodological changes during the study will be updated on the Open Science Framework platform<ref>{{cite web |url=https://osf.io/ym65x/ |title=How to write a scoping review protocol: Guidance and template |last1=Lely |first1=Justine |last2=Morris |first2=Hailey C. |last3=Sasson |first3=Noa |last4=Camarillo |first4=Nathan D. |last5=Livinski |first5=Alicia A. |last6=Butera |first6=Gisela |last7=Wickstrom |first7=Jordan |date=May 30, 2023 |website=OSF |publisher=Center for Open Science |access-date=2023-05-30 |quote= }}</ref>. The search was guided by the following research questions: a) ''"What are the applications and benefits of Wikipedia in universities as a resource for active methodology in the health field?"'' and b) ''"What is Wikipedia's contribution to scientific dissemination in health?"'' The research questions were developed using the Population, Concept, and Context (PCC) strategy: '''P''' - population (Wikipedia in universities and Wikipedia in the health field); '''C''' - concept (contribution, application, and benefits); '''C''' - context (active methodology and scientific dissemination). In the next step, descriptors were identified using [[en:w:Medical Subject Headings|MeSH (Medical Subject Headings)]] and [[en:w:Health Sciences Descriptors|DeCS (Health Sciences Descriptors)]] and structured into the search strategy: '''“Wikipedia” AND (“Wikipedia” AND (“university” OR “health” OR “education”)).''' The article selection process was based on the following inclusion criteria: free full-text articles ([[en:w:Open Access|Open Access]]); articles published between 2015 and 2020; and studies available in Portuguese, French, German, English, and Spanish. Duplicate references were analyzed and excluded. Subsequently, the titles and abstracts of the articles were reviewed based on the criteria and guiding questions. The articles included in the review were then grouped according to the guiding questions defined in the research scope, considering the objective and outcomes of each article. The Rayyan review software ([[en:w:Qatar Computing Research Institute|Qatar Computing Research Institute]]), and the [[en:w:Google Sheets|Google Sheets tool]] for systematizing the review process, with manual verification performed later. A total of 1,318 articles were retrieved from the databases and added to the review software, and a total of 742 articles published between 2015 and 2020 were included, followed by the inclusion of 526 texts in English, Portuguese, and Spanish. This resulted in 386 articles selected for full reading by the authors based on the guiding questions of the literature review. The full reading of the selected studies led to the inclusion of 12 articles, which were then added to the literature review (Figure 1). [[File:Flowchart of the included articles.jpg|center|thumb|410x410px|Flowchart of the selection and inclusion process of articles in the literature review.]] == Results and Discussion == Twelve articles were included in the literature review, with eight articles answering the question ''"What is the application and benefits of Wikipedia in universities as an active learning resource in the health field?"'' and four articles answering the question ''"What is Wikipedia's contribution to scientific dissemination?"''. The categorization of the articles was based on the research questions established, as presented in Table 1 and Table 2, regarding the title, publication year, objective, and outcome of the articles. {| class="wikitable" |+Table 1: Outcomes related to the guiding question: ''What is the application and benefits of Wikipedia in universities as an active learning resource in the health field?'' | | '''Wikipedia Education Program in higher education settings<ref name=":2">{{Cite journal|last=Alcazar|first=Carmen|last2=Bucio|first2=Jackeline|last3=Ferrante|first3=Luisina|date=2018-04-04|title=Wikipedia Education Program in higher education settings: Actions and lessons learned from four specific cases in Mexico and Argentina|url=https://revistas.ucu.edu.uy/index.php/paginasdeeducacion/article/view/1552|journal=Páginas de Educación|volume=11|issue=1|pages=23–36|doi=10.22235/pe.v11i1.1552|issn=1688-7468}}</ref>''' |- |Objective |Present a comparative view of the efforts and results achieved by the Wikipedia Education Program, working with [[en:w:higher education|higher education]] institutions, identifying challenges and similar solutions in implementing Wikipedia projects in academic settings. |- |Outcomes |Feedback was received from the professor and the Wikipedia community. This led to an increase in the number of students who re-edited their [[en:w:homework|assignments]]. Wikipedia can be a resource for critical assessment and content improvement, enabling students to assess [[en:w:consensus|consensus]], generate constructive feedback, and collaborate based on reliable sources. It forms new generations of networked learners. Furthermore, students reported greater motivation to engage in formative assessment activities when their contributions receive measurable exposure, knowing the real and quantifiable audience. |- | colspan="2" | '''Wikipedia as a reference information source: evaluation and perspectives'''<ref name=":3">{{Cite journal|last=Kern|first=Vinícius Medina|date=2018-01|title=A Wikipédia como fonte de informação de referência: avaliação e perspectivas|url=https://doi.org/10.1590/1981-5344/3224|journal=Perspectivas em Ciência da Informação|volume=23|issue=1|pages=120–143|doi=10.1590/1981-5344/3224|issn=1981-5344}}</ref> |- |Objective |What supports the acceptance of Wikipedia in forums such as elite [[en:w:scientific journals|scientific journals]]? |- |Outcomes |Editing Wikipedia is an excellent way to fulfill public participation responsibilities and [[en:w:Knowledge sharing|share knowledge]]. The use of Wikipedia in the university classroom is encouraged by several authors. |- | colspan="2" | '''Uses, perceptions, and evaluations of Wikipedia by university professors'''<ref name=":4">RIVOIR, A. L.; ESCUDER, S.; RODRIGUEZ HORMAECHEA, F. Usos percepciones y valoraciones de Wikipedia por profesores universitarios. '''Innov Educ''' (Méx DF), México, v. 17, n. 75, p. 169-187, dez 2017. </ref> |- |Objective |To understand the uses, perceptions, and evaluations of Wikipedia through a survey of university professors, developing a [[en:w:Typology (social science research method)|typology]] and usage profiles of the tool. |- |Outcomes |There are advantages to using Wikipedia in academic activities at the university, particularly in [[en:w:teaching|teaching]]. As an educational resource, collaborative work stands out. Through article editing, it contributes to [[en:w:writing|writing skills]], analysis, and academic performance. |- | colspan="2" | '''Improving the Quality of Consumer Health Information on Wikipedia: Case Series'''<ref name=":5">{{Cite journal|last=Weiner|first=Shira Schecter|last2=Horbacewicz|first2=Jill|last3=Rasberry|first3=Lane|last4=Bensinger-Brody|first4=Yocheved|date=2019-03-18|title=Improving the Quality of Consumer Health Information on Wikipedia: Case Series|url=https://doi.org/10.2196/12450|journal=Journal of Medical Internet Research|volume=21|issue=3|pages=e12450|doi=10.2196/12450|issn=1438-8871}}</ref> |- |Objective |Enhance Wikipedia's health pages using current high-quality research findings and track the persistence of these edits and the number of page views after the changes to evaluate the initiative's impact. |- |Outcomes |Taking on the role of Wikipedia editor required students to gather and synthesize current, [[en:w:Evidence-based practice|evidence-based]] information, promoting learning. |- | colspan="2" | '''Pharmacy students can improve access to quality medicines information by editing Wikipedia articles'''<ref name=":6">{{Cite journal|last=Apollonio|first=Dorie E.|last2=Broyde|first2=Keren|last3=Azzam|first3=Amin|last4=De Guia|first4=Michael|last5=Heilman|first5=James|last6=Brock|first6=Tina|date=2018-12|title=Pharmacy students can improve access to quality medicines information by editing Wikipedia articles|url=https://bmcmededuc.biomedcentral.com/articles/10.1186/s12909-018-1375-z|journal=BMC Medical Education|language=en|volume=18|issue=1|doi=10.1186/s12909-018-1375-z|issn=1472-6920}}</ref> |- |Objective |Expand the traditional approach of [[en:w:pharmacy|pharmacy training programs]] by requiring students to improve [[en:w:Medication|medication information]] pages on Wikipedia. |- |Outcomes |Editing Wikipedia allows students to demonstrate their skills in research and in conveying information about medications. Through this experience, students come to view the platform in a more informed and critical way. |- | colspan="2" | '''Wikipedia as a gateway to biomedical research: The relative distribution and use of citations in the English Wikipedia'''<ref name=":7">{{Cite journal|last=Maggio|first=Lauren A.|last2=Willinsky|first2=John M.|last3=Steinberg|first3=Ryan M.|last4=Mietchen|first4=Daniel|last5=Wass|first5=Joseph L.|last6=Dong|first6=Ting|date=2017-12-21|editor-last=Sugimoto|editor-first=Cassidy Rose|title=Wikipedia as a gateway to biomedical research: The relative distribution and use of citations in the English Wikipedia|url=https://dx.plos.org/10.1371/journal.pone.0190046|journal=PLOS ONE|language=en|volume=12|issue=12|pages=e0190046|doi=10.1371/journal.pone.0190046|issn=1932-6203}}</ref> |- |Objective |Establish benchmark parameters for the relative distribution and click-through rate of [[en:w:citations|citations]]—as indicated by the presence of a [[en:w:Digital Object Identifier|Digital Object Identifier (DOI)]]—on Wikipedia, with a focus on medical citations. |- |Outcomes |The introduction of continuous teaching strategies in undergraduate and medical and health education should address the value of consulting citations as a tool to enhance learning. For example, at the [[en:w:University of California|University of California]], San Francisco, the School of Medicine offers training for students to edit and critically evaluate Wikipedia. |- | colspan="2" | '''The 5th ISCB Wikipedia Competition: Coming to a Classroom Near You?'''<ref name=":8">{{Cite journal|last=Kilpatrick|first=Alastair M.|date=2016-12-29|title=The 5th ISCB Wikipedia Competition: Coming to a Classroom Near You?|url=https://doi.org/10.1371/journal.pcbi.1005235|journal=PLOS Computational Biology|volume=12|issue=12|pages=e1005235|doi=10.1371/journal.pcbi.1005235|issn=1553-7358}}</ref> |- |Objective |Show the [[en:w:International Society for Computational Biology|5th International Society for Computational Biology]] competition on Wikipedia. |- |Outcomes |The use of competitions in courses, involving contributions to Wikipedia in the [[en:w:classroom|classroom]], was considered a success. Unlike traditional classes, by contributing to Wikipedia, students' work becomes available to the public and future researchers. The competition also provides an opportunity to develop effective communication skills with diverse audiences and to work effectively as part of a team. |- | colspan="2" | '''Why Medical Schools Should Embrace Wikipedia: Final-Year Medical Student Contributions to Wikipedia Articles for Academic Credit at One School'''<ref name=":9">{{Cite journal|last=Azzam|first=Amin|last2=Bresler|first2=David|last3=Leon|first3=Armando|last4=Maggio|first4=Lauren|last5=Whitaker|first5=Evans|last6=Heilman|first6=James|last7=Orlowitz|first7=Jake|last8=Swisher|first8=Valerie|last9=Rasberry|first9=Lane|date=2017-02|title=Why Medical Schools Should Embrace Wikipedia: Final-Year Medical Student Contributions to Wikipedia Articles for Academic Credit at One School|url=https://journals.lww.com/00001888-201702000-00022|journal=Academic Medicine|language=en|volume=92|issue=2|pages=194–200|doi=10.1097/ACM.0000000000001381|issn=1040-2446|pmc=PMC5265689|pmid=27627633}}</ref> |- |Objective |Assess the impact of edits made by fourth-year medical students on Wikipedia content, as well as the effects of the course on both the participants and the readers of the articles selected by the students. |- |Outcomes |The students reported challenges in addressing the general public and the professional medical audience. They found the collaborative nature of the work to be a challenge, as it often required a willingness to re-edit; however, these challenges were mitigated by the perception of the usefulness and potential global significance of their contributions. Additionally, they developed their skills as health educators and completed the course with an expanded understanding of their professional responsibilities. They gained practical experience in explaining health information in an accessible manner to the general public, which they found rewarding. |} {| class="wikitable" |+Table 2: Outcomes related to the guiding question: ''What is Wikipedia's contribution to scientific dissemination?'' ! colspan="2" |The Most Influential Medical Journals According to Wikipedia: Quantitative Analysis<ref name=":10">{{Cite journal|last=Jemielniak|first=Dariusz|last2=Masukume|first2=Gwinyai|last3=Wilamowski|first3=Maciej|date=2019-01-18|title=The Most Influential Medical Journals According to Wikipedia: Quantitative Analysis|url=http://www.jmir.org/2019/1/e11429/|journal=Journal of Medical Internet Research|language=en|volume=21|issue=1|pages=e11429|doi=10.2196/11429|issn=1438-8871}}</ref> |- |Objective |Determine the ranking of the most cited journals based on their representation in English-language medical pages on Wikipedia. Assess the number of days between the publication of journal articles and their citation on the platform ([[en:w:information access|information dissemination]]). |- |Outcomes |The time between the publication of a journal article and its mention on Wikipedia has decreased substantially since the platform's inception. Evidence of "[[en:w:presentism|presentism]]" (a preference for citing recently published articles) was found. High-impact traditional medical and [[en:w:multidisciplinary|multidisciplinary]] journals were highly cited by Wikipedia, suggesting that Wikipedia's medical articles have solid foundations. This allows the inclusion of general science and health journals and serves as an alternative, if not more reliable, measure of a journal's impact on public knowledge, based on decisions made by the [[en:w:Self-governance|self-governing]], peer-production community. |- | colspan="2" |'''Wikipedia and medicine: quantifying readership, editors, and the significance of natural language'''<ref>{{Cite journal|last=Heilman|first=James M|last2=West|first2=Andrew G|date=2015-03-04|title=Wikipedia and Medicine: Quantifying Readership, Editors, and the Significance of Natural Language|url=http://www.jmir.org/2015/3/e62/|journal=Journal of Medical Internet Research|language=en|volume=17|issue=3|pages=e62|doi=10.2196/jmir.4069|issn=1438-8871}}</ref> |- |Objective |Quantify the production and consumption of medical content on Wikipedia across four dimensions: 1. Articles and bytes; 2. Citations supporting the content; 3. Analysis of medical readers compared to other health websites; 4. Characteristics of medical contributors to Wikipedia. |- |Outcomes |By the end of 2013, Wikipedia's medical content encompassed over 155,000 articles and 1 billion bytes of text in more than 255 languages (or 1016 MB), representing a 10.19% increase from the previous year (922 MB). The content was supported by over 950,000 references from journals respected by the scientific community. Health articles were viewed more than 4.9 billion times annually on non-mobile devices (with inclusive estimates for [[en:w:mobile devices|mobile devices]] reaching 6.5 billion). This makes it one of the most viewed medical resources in the world. Since 2010, the number of academic articles in health sciences citing Wikipedia has increased substantially. |- | colspan="2" | '''Improving the Quality of Consumer Health Information on Wikipedia: Case Series'''<ref name=":5" /> |- |Objective |To improve Wikipedia health pages by incorporating high-quality, up-to-date research findings and monitoring the persistence of these edits along with the number of page views to evaluate the initiative's impact. |- |Outcomes |The reach and potential of the Internet to improve the health of individuals and populations should be recognized by healthcare professionals as a tool for enhancing the quality and [[en:w:equity|equity]] of [[en:w:healthcare services|healthcare services]]. Wikipedia facilitates the dissemination of a large volume of high-quality information to broad segments of the population. Wikipedia's standards require the inclusion of citations, a practice that other health information sources on the internet do not always follow. The Wikipedia model can streamline the translation of research into practice. |- | colspan="2" | '''Pharmacy students can improve access to quality medicines information by editing Wikipedia articles'''<ref name=":6" /> |- |Objective |Expand the traditional approach of pharmacy training programs by engaging students in improving medication information pages published on Wikipedia. |- |Outcomes |Wikipedia is frequently used by students, healthcare professionals, and patients to quickly access information and enables the sharing of [[en:w:medicine|medical knowledge]] in a way that is accessible to the public. Encouraging students to improve health content on Wikipedia not only benefits the users of the pages but also helps the students develop skills in communicating information to the public. |} === Application and benefits of Wikipedia in universities as an active learning resource in the health field === There is agreement that Wikipedia can be used in universities as an active learning resource in the health field through student edits to entries. This strategy allows for the development of [[w:en:research|research skills]]<ref name=":6" /> <ref name=":4" />, writing, analysis, and improved academic performance<ref name=":4" /> through collaborative work as part of a team<ref name=":8" />, and by exploring the practice of consensus<ref name=":2" />. Furthermore, this exercise can introduce students to or expand their experience with the practice of peer review in an amplified, public review process<ref name=":11">{{Cite journal|last=Cummings|first=Robert E.|date=2020-04-21|title=Writing knowledge: Wikipedia, public review, and peer review|url=https://doi.org/10.1080/03075079.2020.1749791|journal=Studies in Higher Education|volume=45|issue=5|pages=950–962|doi=10.1080/03075079.2020.1749791|issn=0307-5079}}</ref>. The work encourages the collection and synthesis of the most relevant current information for subsequent editing, thereby promoting learning on the subject<ref name=":5" /> . The platform's updating requires critical evaluation in the selection of sources, data, and changes to previous edits<ref name=":2" /> <ref name=":7" />. This also fosters a sense of agency<ref name=":2" />, where greater motivation is achieved through fulfilling [[en:w:social participation|social participation]] responsibilities<ref name=":3" />, coupled with the sense of utility and importance provided by [[en:w:Knowledge sharing|sharing knowledge]] with communities lacking access to other validated [[en:w:information|sources of information]]<ref>{{Cite journal|last=Azzam|first=Amin|last2=Bresler|first2=David|last3=Leon|first3=Armando|last4=Maggio|first4=Lauren|last5=Whitaker|first5=Evans|last6=Heilman|first6=James|last7=Orlowitz|first7=Jake|last8=Swisher|first8=Valerie|last9=Rasberry|first9=Lane|date=2017-02|title=Why Medical Schools Should Embrace Wikipedia: Final-Year Medical Student Contributions to Wikipedia Articles for Academic Credit at One School|url=https://journals.lww.com/00001888-201702000-00022|journal=Academic Medicine|language=en|volume=92|issue=2|pages=194–200|doi=10.1097/ACM.0000000000001381|issn=1040-2446}}</ref>. The gratification reported by students in this and other formative assessment activities is attributed to measurable reach to a real audience<ref name=":2" /><ref name=":9" />. Not only, does the professor provide feedback to students, but also the entire Wikipedia community, leading to expanded feedback that encourages constant revisions of their work <ref name=":2" /><ref name=":11" />. Additionally, from concrete experiences, there is effective development of communication skills with diverse groups, ranging from the general population to the medical/professional community<ref name=":8" />. In this context, editing allows students to explore and develop skills as health educators, gaining an enhanced understanding of their professional roles<ref name=":9" />. Unlike traditional classes, encouraging contributions to Wikipedia means advancing in favor of current modernity<ref name=":3" /> and creating active, engaged, networked learners <ref name=":2" />. === Contribution of Wikipedia to scientific dissemination === The literature has highlighted the extensive reach and potential of Wikipedia as one of the top five online information sources<ref name=":5" />. The medical content available on Wikipedia grew by 10.2% from 2012 to the end of 2013, with over 155,000 articles, 1 billion bytes of textual content in more than 255 languages, supported by more than 950,000 references, and over 4.9 billion views annually from computers—estimated to reach 6.5 billion when including mobile devices. As such, Wikipedia has become one of the most accessed medical resources worldwide<ref name=":1" />. Increasingly, traditional scientific journals formally reference the platform<ref name=":10" />, and since 2010, there has been a significant rise in citations of the encyclopedia in academic articles within the health sciences field. Additionally, the reverse is observed, as Wikipedia uses references from the most respected journals <ref>{{Cite journal|last=Heilman|first=James M|last2=West|first2=Andrew G|date=2015-03-04|title=Wikipedia and Medicine: Quantifying Readership, Editors, and the Significance of Natural Language|url=http://www.jmir.org/2015/3/e62/|journal=Journal of Medical Internet Research|language=en|volume=17|issue=3|pages=e62|doi=10.2196/jmir.4069|issn=1438-8871}}</ref>. The distinctiveness of Wikipedia compared to most other available online health information sources lies in its requirement for the inclusion of citations<ref name=":5" />. These citations from high-impact medical and multidisciplinary journals also demonstrate a considerable reduction in the time between the publication of an article and its inclusion on Wikipedia<ref name=":10" />. Given its broad grounding, these factors suggest that health articles in the encyclopedia correspond to quality content with solid foundations. Furthermore, they constitute an important resource for enhancing the visibility of major journals in the public domain through peer production<ref name=":10" />. === Recognition by the scientific community === The recognition of Wikipedia’s contribution to knowledge dissemination can also be observed through its inclusion in the analysis of new [[en:w:Bibliometrics|bibliometric indicators]], which aim to address the gaps of the traditional "impact factor." The traditional impact factor has several limitations, one of which is that articles published in high-impact journals but receiving few citations benefit from the quality of more cited articles, potentially leading to a [[en:w:misjudgment|misjudgment]] of the relevance and quality of the work. New metrics measure the individual impact of a scientific article through the number of citations that encompass [[en:w:social networks|social networks]] in their search, in addition to accounting for traditional sources<ref>{{Cite journal|last=VANTI|first=Nadia|last2=SANZ-CASADO|first2=Elias|date=2016-12|title=Altmetria: a métrica social a serviço de uma ciência mais democrática|url=https://doi.org/10.1590/2318-08892016000300009|journal=Transinformação|volume=28|issue=3|pages=349–358|doi=10.1590/2318-08892016000300009|issn=2318-0889}}</ref>. The mention of an article in a Wikipedia entry is included and counted in the impact assessment of new systems. These systems allow researchers to discover the reach of studies on the web, at a time when increasingly, these informal social media platforms facilitate and accelerate information exchange in the academic world. It is important to note, however, that the quality and relevance of the coverage of scientific topics in Wikipedia entries may be directly or indirectly influenced by the level of involvement of area specialists. The expansion and improvement of articles in a given field or subject generally lead to an increase in views<ref name=":1" />. Retaining teachers and students as active collaborators on the platform could solidify this mechanism of scientific dissemination. [[en:w:WikiProject|WikiProjects]] exist to facilitate contact between members of the existing community with common interests, increasing the adoption rate of new collaborators. The English Wikipedia contains nearly 2,000 WikiProjects (around 800 active), and approximately 30 of them are in the field of health<ref>{{Cite journal|date=2024-06-03|title=Wikipedia:WikiProject Lists|url=https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Lists|journal=Wikipedia|language=en}}</ref>. In English Wikipedia, the articles that are part of the [[Medicine]] WikiProject (the largest WikiProject in the health field) are generally more comprehensive, have a greater number of references, a higher [[en:w:citation index|citation index]], and receive more visits when compared to articles in the rest of Wikipedia. Furthermore, readers of these pages tend to show higher engagement with the content when compared to readers of other articles, spending more time on the pages before clicking on references and checking footnotes more frequently<ref>{{Cite journal|last=Maggio|first=Lauren A|last2=Steinberg|first2=Ryan M|last3=Piccardi|first3=Tiziano|last4=Willinsky|first4=John M|date=2020-03-06|title=Reader engagement with medical content on Wikipedia|url=https://elifesciences.org/articles/52426|journal=eLife|language=en|volume=9|doi=10.7554/eLife.52426|issn=2050-084X}}</ref>. Thus, it is understood that Wikipedia is a strong ally in the dissemination of a vast amount of high-level knowledge to substantial segments of the population, with current and meaningful information for the public<ref name=":5" />. In fact, encouraging students to improve content on Wikipedia not only benefits them individually but also contributes to the broad scientific dissemination to society<ref name=":6" />. ==Conclusion== The reviewed literature allowed for answering the guiding questions of this study, demonstrating benefits for the development of academic, professional, and personal skills resulting from the incorporation of Wikipedia as an active and participatory methodological resource in health courses. Thus, these findings suggest the encouragement of editing practice on the platform, aiming at expanding learning and fulfilling social responsibilities in health education or communication. Consequently, these activities contribute to making one of the most read and accessible sources of information in the world even more accurate and comprehensive in the field, disseminating relevant knowledge supported by scientific publications to millions of people. '''Acknowledgements''' Acknowledgment to the [[en:w:São Paulo Research Foundation|São Paulo Research Foundation]] (FAPESP) for funding grant No. [https://bv.fapesp.br/en/auxilios/109545/wikipedia-education-program-as-active-teaching-methodology-and-crowdsourcing-tool-in-hearing-health/ 2021/06902-2] - Wikipedia Education Program as active teaching methodology and crowdsourcing tool in hearing health. ===Competing interests=== The authors declare no conflict of interest. ==References== {{reflist|35em}} [[Category:Wikipedia]] to1muue01lsqj4hpe4eqzuisv2hw00t 2 317046 2691713 2691554 2024-12-12T23:06:29Z Jaredscribe 2906761 2691713 wikitext text/x-wiki {{proposed policy}} A "table of contents" is a "selection and arrangement" (and as such may be copyrighted). This essay argues that the reproduction of a book's table of contents on wikiversity is fair use, when it is done for the sake of making a "book report" or summarizing its contents, as is done on article on [[w:Harry_Potter_and_the_Philosopher's_Stone#Podcast_version|Harry Potter and the Philosopher's Stone]], a fictional book, lists its table of contents: all the more so for a book of history, science, or philosophy. And even more soso for a work of science or philosophy that is itself a derivative or works in the public domain. * [[w:Help:Beginner's_guide_to_copyright|Help:Beginner's guide to copyright]] * [[Wikiversity:Uploading files#Fair use considerations]] * [[Wikiversity:Academic freedom]] * [[Wikiversity:Copyrights]] * [[Wikiversity:Copyright issues]] * [[Meta:Avoid copyright paranoia]] * The article on [[w:Harry_Potter_and_the_Philosopher's_Stone#Podcast_version|Harry Potter and the Philosopher's Stone]], a fictional book, lists its table of contents. * [[Wikivesity:Book report]] * [[Wikiversity:Original historical research]] * [[Wikiversity:Original research in current events]] * [[Wikiversity:Original criticism]] commentaries on notable works of philosophy, literature, and religion that may or may not yet be in the public domain. * we encourage use of [[w:Wikipedia:Primary sources]], and unlike wikipedia, allow [[w:Template:AEIS]], assuming that scholarly ethics and intellectual honesty are in practice. * Adler's is a [[w:Wikipedia:FAQ/Copyright#Derivative_works]] of public domain. * Uncertain whether his "second table of contents" is [[w:Free_content|free content]] or [[w:Wikipedia:Non-free_content|non-free content]]. * If non-free, it may be permissible as "fair use" [[w:Wikipedia:FAQ/Copyright#What_is_fair_use?]] * [[w:Wikipedia:Copyrights#Using_copyrighted_work_from_others]] * [[w:Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing|Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing]] * [[w:User:Jaredscribe/WV:Toc_is_fair_use|User:Jaredscribe/WV:Toc is fair use]] * [[Wikipedia:Copyrights#Re-use_of_text]] *[[Wikipedia:FAQ/Copyright#What_is_fair_use?]] : Under US copyright law, the primary things to consider when asking if something is fair use (set forth in Title 17, Chapter 1, Section 107) are: :# The purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes; :# The nature of the copyrighted work; :# The amount and substantiality of the portion used in relation to the copyrighted work as a whole; and :# The effect of the use upon the potential market for or value of the copyrighted work. : Asking yourself these questions might help you determine if something is fair use: :# ''Is it a for profit competitor or not? Is it for criticism, comment, news reporting, teaching, scholarship, or research? Is the use transformative (of a different nature to the original publication)?'' :# ''Is it a highly original creative work with lots of novel ideas or a relatively unoriginal work or listing of facts? Is the work published (to a non-restricted audience)?'' If not, fair use is much less likely. :# ''How much of the original work are you copying? Does the portion that you are copying constitute the "heart" of the work and/or its most powerful and significant part? Are you copying more or less than the minimum required for your purpose?'' The more you exceed this minimum, the less likely the use is to be fair. ''Are you reducing the quality or originality, perhaps by using a reduced size version?'' :# ''Does this use hurt or help the original author's ability to sell it; in particular, does it replace the market for authorized copies? Did they intend to or were they trying to make the work widely republished (as with a press release)? Are you making it easy to find and buy the work if a viewer is interested in doing so?'' If non-free, and if under some legal theory this is not allowed under fair use, then requesting the publisher: * [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] * [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] * [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. lrr6rftykm7xofjuf7aehoufq13eutr 0 317067 2691632 2691516 2024-12-12T14:02:52Z Eshaa2024 2993595 2691632 wikitext text/x-wiki == Definition: Path == Let <math>U \subset \mathbb{C}</math> be a subset. A path in <math>U</math> is a [[w:en:continuous function|continuous mapping]] with: :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. == Definition: Trace of a Path == The trace of a path <math>\gamma \colon [a,b] \rightarrow U </math> in <math>U \subset \mathbb{C}</math> is the [[w:en:Range of a function|image or range]] of the function <math>\gamma</math>: :<math>\mathrm{{Trace}}(\gamma):= { \gamma(t) \in \mathbb{C} \ | \ t \in [a,b] }</math> == Definition: Closed Path == Let <math>\gamma \colon [a,b] \rightarrow U </math> be a path in <math>U \subset \mathbb{C}</math>. The mapping <math>\gamma</math> is called a closed path if: :<math>\gamma(a) = \gamma(b)</math> == Definition: Region == Let <math>U \subset \mathbb{C}</math> be an open subset of <math>\mathbb{C}</math>. Then <math>U</math> is called a region. == Definition: Path-Connected == Let <math>U \subset \mathbb{C}</math> be a non-empty set. :<math>U</math> is path-connected <math>:\Longleftrightarrow \ \forall_{z_1,z_2 \in U }\exists_{\gamma\colon [a,b]\rightarrow U}: \ \gamma(a)=z_1 \wedge \gamma(b)=z_2 \wedge{{Spur}}(\gamma) \subseteq U</math> == Definition: Domain == Let <math>G \subset \mathbb{C}</math> be a non-empty subset of <math>\mathbb{C}</math>. If *<math>G</math> is open *<math>G</math> is path-connected Then <math>G</math> is called a domain in <math>\mathbb{C}</math>. == Example (Circular Paths) == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>r > 0</math> be a radius. A circular path <math>\gamma_{z_o,r}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,r}(t):= z_o + r\cdot e^{i\cdot t}</math> == Example - Paths with Ellipse as Trace == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>a, b > 0</math> be the semi-axes of an ellipse. An elliptical path <math>\gamma_{z_o,a,b}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,a,b}(t):= z_o + a\cdot \cos(t) + i\cdot b\cdot \sin(t)</math> == Gardener's Construction of an Ellipse == [[File:Elliko-g.svg|350px|Gardener's Construction of an Ellipse]] == Convex Combinations == Let <math>z_1,z_2 \in \mathbb{C}</math> be complex numbers, and let <math>t \in [0,1]</math> be a scalar. A path <math>\gamma_{z_1,z_2}\colon [0,1] \rightarrow \mathbb{C}</math> is defined such that its trace is the line segment connecting <math>z_1,z_2 \in \mathbb{C}</math>: :<math>\gamma_{z_1,z2}(t):= (1-t)\cdot z_1 + t\cdot z_2</math> Such a path is called a convex combination of the first order (see also [[Convex Combination|Convex Combinations of higher order]]). === Animation of a Convex Combination of Two Vectors as Mapping === [[File:Convex combination 1 ord with geogebra.gif|450px|center|Convex Combination as Mapping in an Animated GIF]] == Integration Path == Let <math>G \subset \mathbb{C}</math> be a domain. An integration path in <math>G</math> is a path that is piecewise continuously differentiable with :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. === Remark === An integration path can, for example, be expressed piecewise as convex combinations between multiple points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The overall path does not need to be differentiable at points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The trace of such a path is also called a polygonal path. == See Also == * [[w:en:Ellipse|Ellipse]] * [[Convex Combination]] * [[Topological vector space]] == Page Information == You can display this learning resource as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Wege&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wege&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Wege&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wege&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/_Kurs:Funktionentheorie Course:Complex Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. <!-- * The content of the page is based on the following material: ** [https://de.wikipedia.org/wiki/Kurs:Funktionentheorie/Wege https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Wege] --> [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Wege The page] was created as a document type [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Wege See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Kurs:Funktionentheorie/Wege&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wege&coursetitle=Kurs:Funktionentheorie Wiki2Reveal link generator]. <!-- * The next content of the course is [[]] -->[[Category:Wiki2Reveal]] 9q0yto0st0ln3g8cb1dzxmaja87vivh 0 317068 2691628 2691371 2024-12-12T13:48:27Z Eshaa2024 2993595 2691628 wikitext text/x-wiki == Introduction == This page on the topic "Path Integral" can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Pathintegral&author=Course:Functiontheory&language=de&audioslide=yes&shorttitle=Pathintegral&coursetitle=Course:Functiontheory Wiki2Reveal Slides]'''. Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides. The following subtopics are treated in detail: (1) Paths as continuous mappings from an interval <math>[a,b]</math> into the complex numbers <math>\mathbb{C}</math> over which integration is performed, (2) Derivatives of curves/paths as a prerequisite for the definition of path integrals, (3) Definition of path integrals == Learning requirements == The learning resource on the topic "Path Integral" has the following learning prerequisites, which are helpful or necessary for understanding the subsequent explanations: *Concept of[[w:en:Paths in a topological Space|Paths in a topological Space]], *Differentiability in real analysis, *Integration in real analysis. == Basic Geometric Idea of the Path Integral == The following curve <math>\gamma</math> loops around a point <math>z_0\in \mathbb{C}</math> twice. [[File:Windungszahl5.png|150px|center|Path around a point]] == Integral over an Interval == Let <math>G\subseteq \mathbb{C}</math> be a [[w:en:Domain (mathematics)|domain]] and <math>g\colon [a,b] \to\mathbb{C}</math> a [[w:en:Complex function|complex-valued function]]. The function <math>g</math> is called integrable if ::<math>\operatorname{Re}(g):G \to\mathbb{R}</math> and <math>\operatorname{Im}(g):G \to\mathbb{R}</math> with <math>g=\operatorname{Re}(g) + i \cdot \operatorname{Im}(g)</math> are integrable functions. It is defined as :<math>\int\limits_a^b g(x)\mathrm{d} x := \int\limits_a^b\operatorname{Re}(g)(x)\mathrm{d}x +\mathrm{i}\int\limits_a^b\operatorname{Im}(g)(x)\mathrm{d}x</math>. Thus, the integral is <math>\mathbb{C}</math>-linear. If <math>g</math> is continuous and <math>G</math> is an antiderivative of <math>g</math>, then as in the real case, :<math>\int\limits_a^b g(x)\mathrm{d}x = G(b)-G(a)</math>. == Extension of the Integral Concept == The integral concept is extended through the definition of an integration path in the complex plane as follows: If <math>f\colon G\to\mathbb{C}</math> is a complex-valued function on a [[w:en:Domain (mathematics)|domain]] <math>G\subseteq\mathbb{C}</math>, and <math>\gamma\colon[a,b]\to G</math> is a piecewise continuously differentiable [[w:en:Path (mathematics)|path]] in <math>G</math>, then the ''path integral'' of <math>f</math> along the path <math>\gamma</math> is defined as : <math>\int\limits_\gamma f:=\int\limits_\gamma f(z),\mathrm dz:=\int\limits_a^b f(\gamma(t))\cdot \gamma'(t),\mathrm dt.</math> Here, the multiplication sign refers to complex multiplication.<ref>„Curve Integral“. In: Wikipedia, The Free Encyclopedia. Editing status: November 24, 2017, 16:22 UTC. URL: https://en.wikipedia.org/w/index.php?title=Curve_integral&oldid=171345033 (Accessed: December 8, 2017, 14:27 UTC) </ref> == Cauchy's Integral Theorem == The central statement about path integrals of complex functions is the [[w:en:Cauchy's_integral_theorem|Cauchy Integral Theorem]]: For a [[w:en:Holomorphic function|holomorphic]] function <math>f</math>, the path integral depends only on the [[w:en:Homotopy|homotopy]] class of <math>\gamma</math>. If <math>U</math> is [[w:en:Simply_connected_space|simply connected]], then the integral depends not on <math>\gamma</math>, but only on the starting and ending points. Analogous to the real case, the ''length'' of the path <math>\gamma:[a,b]\rightarrow \mathbb{C}</math> is defined as :<math>\operatorname{L}(\gamma):=\int\limits_a^b \left| \gamma'(t) \right| \mathrm{d}t</math>. For theoretical purposes, the following inequality, called the ''standard estimate'', is of particular interest: :<math>\left| \int_\gamma f(z) , \mathrm dz \right| \leq \operatorname{L}(\gamma)\cdot C</math>, if <math>\left| f(z) \right|\leq C</math> for all <math>z\in\gamma([0,1])</math>. As in the real case, the path integral is independent of the parametrization of the path <math>\gamma</math>, i.e., it is not strictly necessary to choose <math>[0,1]</math> as the parameter domain, as can be shown by substitution. This allows the definition of complex curve integrals by replacing the above formulas with a curve <math>\mathcal{C}</math> in <math>\mathbb{C}</math>. == Exercises == *Be <math>\gamma\colon[a,b]\to G</math> with <math>t\mapsto \gamma(t)= \sin(t)+i\cdot t^2</math>. Determine <math>\gamma'(t)</math>! *Compute the path integral <math>\int\limits_\gamma \frac{1} {z},\mathrm dz</math> for the path <math>\gamma\colon[0,2\pi] \to \mathbb{C}</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. *Calculate the length of the path <math>L(\gamma)</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. == See also == *[[w:en: Function theory (course)|Function Theory Course]] *[[w:en: Curve integral|Curve integral]] == Literature == <references/> == Page Information == You can display this learning resource as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie Course:Complex Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. <!-- * The content of the page is based on the following material: ** [https://de.wikipedia.org/wiki/Wegintegral https://de.wikiversity.org/wiki/Wegintegral] -->[https://de.wikiversity.org/wiki/Wegintegral The page] was created as a document type [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/Wegintegral See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/index.html? bceovtzupop84u988x14qbsisnhy640 2 317112 2691749 2691313 2024-12-13T07:38:22Z Jaredscribe 2906761 /* 5784 - 2024 in context */ 2691749 wikitext text/x-wiki This is reading list and study notes. resume is on [[User:Jaredscribe#Zionism, Palestine, Israel]] The [[Arab%E2%80%93Israeli alliance against Iran]] emerged in 2017. The [[February 2019 Warsaw Conference]] prepared the way for the [[Abraham Accords]], the [[Israel%E2%80%93United Arab Emirates normalization agreement]], and the {{slink|Arab%E2%80%93Israeli conflict#Israeli normalization with Gulf states and Sudan}} [[Negev Summit]] Recent phase of the [[Arab League and the Arab%E2%80%93Israeli conflict]] * 5 Iyar [[Independence Day (Israel)]], 15 May [[Nakba Day]] * 30 Nov [[Day to Mark the Departure and Expulsion of Jews from the Arab Countries and Iran]] * Libyan [[Day_of_Revenge]]/Friendship? [[Proto-Zionism]] == [[Mandatory Palestine]] == {{blockquote| text=We have watched the rise and fall of this Palestine problem. In 1917 Jewish hopes were raised in all parts of the world. It was thought that at long last here was the Jewish Magna Carta. By 1921 Transjordan was lopped off; in 1922 free immigration became immigration on the basis of absorptive capacity; in 1933 land sales were restricted; in 1937, partition was accepted by the Government; and in 1938 partition was rejected by the Government; and in 1939 we see the funeral of the mandate."| author=[[Tom Williams, Baron Williams of Barnburgh|Tom Williams]] was member of Churchhill's [[Coalition Government 1940-1945]], and later of the government of [[Clement Attlee]] }} [[Haganah|Hagana]] [[Jewish Supernumerary Police|Supernumerary police]] [[Orde Wingate]] [[Havlagah|Havlaga Policy of Restraint]] [[Irgun]] [[Lehi (militant group)|Lehi]] Stern Dr. [[Chaim Weizmann]] [[Peel Commission]] [[Woodhead Commission]] 17 May [[White Paper of 1939|1939 White Paper]] 21st [[World Zionist Congress|Zionist Congress]] 28 Feb 1940 British gov't issued the land transfer regulations envisioned in the White Paper: Palestine divided into Zone A (only to Palestinian Arab), Zone B (Gov't discretion), Zone C (no restrictions). [[Jewish National Fund]]. == Israeli Independence and Arab Catastrophe == [[London Conference of 1946–47]] boycotted except by Arab League states, who argue against partition. [[United Nations Special Committee on Palestine|UNSCOP]] majority proposal 8/11, Zionist side accepts, Arab side rejects. General Assembly appoints [[Ad Hoc Committee on the Palestinian Question]] which .. 29 Nov 1947 [[United Nations Partition Plan for Palestine]], Resolution 181. Accepted by the [[Jewish Agency for Israel|Jewish Agency]], rejected by Arab leaders. [[1947–1948 civil war in Mandatory Palestine]] breaks out. [[1947–1949 Palestine war]] duplicate? [[SS Exodus|SS Exodus 1947]] [[Arab Higher Committee]]: [[Grand Mufti of Jerusalem]] and president of [[Supreme Muslim Council]] [[Amin al-Husayni]], [[Raghib al-Nashashibi]] [[Plan Dalet]] March 1948 arms smuggling [[Operation Balak]] [[Va'ad Leumi]], [[Jewish Agency for Israel|Jewish Agency]], Haganah Command, prominent figures from trade unions, religion, literature and arts, representatives of [[Yishuv|the Yishuv]], at the [[Tel Aviv Museum|Tel Aviv Museum hall]] 5 Iyar / 14 May [[David Ben-Gurion]] proclaims [[Israeli Declaration of Independence]], [[s:Declaration of Independence (Israel)]] [[Bevingrad]] British general [[Alan Cunningham]] Levett Machal [[Sar-El|Sar-el]] [[1948 Arab–Israeli War]] == Divided 1951-1967 == [[King Abdullah of Jordan|King Abdullah]] crowned [[king of Jerusalem]] by the Coptic bishop. He had himself declared king of Palestine in Jericho, and renamed the realm "[[Kingdom of Jordan|United Kingdom of Jordan]]".{{Sfn|Montefiore|2011|p=504}} Appointed [[Raghib al-Nashashibi]] governer of West Bank and Custodian of the two harams (Jerusalem and Hebron) Appointed Sheikh [[Husam al-Jarallah]] as mufti. [[Farouk of Egypt|King Farouk]] overthrown by [[Free Officers Movement (Egypt)|Free Officers of Egypt]], led by general [[Mohamed Naguib|Muhammad Neguib]] and colonel [[Gamal Abdel Nasser|Gamel Abdel Nasser]]. == Six Days and Jerusalem == , [[United Arab Republic]] [[Straits of Tiran]] blockade, [[Israeli passage through the Suez Canal and Straits of Tiran]]. [[Six day war]] [[Egypt–Israel peace treaty]], [[Israel–Jordan peace treaty]], [[Israel–Saudi Arabia relations#Straits of Tiran]] ==5782&mdash;2022== Blinken and Lapid would head to Sde Boker for the [[w:Negev_Summit|Negev Summit]], along with their Bahraini, Egyptian, Emirati and Moroccan counterparts. Blinken called the Houthis’ attack on Iran “acts of terrorism enabled by Iran.” Blinken also pointed out that Iran continues to engage in aggressive actions throughout the Middle East and beyond, directly and through proxies, and spoke out against “mounting terrorist attacks by the Houthis on civilians and civilian infrastructure in the UAE and Saudi Arabia.” “The US will continue to stand up to Iran when it threatens us or our allies and partners,” Blinken said. The US believes “the [[w:JCPOA|JCPOA]] is the best way to put Iran back in the box,” Blinken said, referring to the 2015 nuclear deal to which the US seeks to return, and Israel opposes. Regardless of whether that effort succeeds, he said, “our commitment to the core principle that Iran can never acquire a nuclear weapon is unwavering.” “One way or another, we will continue to cooperate closely,” he stated. “Iran is not an Israeli problem,” Lapid said. “The whole world cannot allow there to be a nuclear Iran and cannot allow the Iranian Revolutionary Guard Corps to continue to spread terror across the globe.” Lapid: “Israel will do everything it thinks is right in order to stop the Iran nuclear program. Everything.” Lapid: “That is not a theoretical threat for us,” he added. “The Iranians want to destroy Israel. They won’t succeed. We won’t let them.” Blinken, US secretary of state: "[the Negev Summit ] would have been unthinkable just a few years ago,” and said the US is committed to expanding cooperation through the Abraham Accords. Blinken: "[will] affirm America’s ironclad commitment to Israel’s security.” Biden signed the omnibus funding bill that included $1 billion for the Iron Dome missile defense system. [https://www.jpost.com/israel-news/politics-and-diplomacy/article-702431 JPost Negev Summit] [https://www.nytimes.com/2022/03/25/world/middleeast/israel-uae-bahrain-morocco-arab-summit.html NYTimes Negev Summit] 5 August 2022 [[Operation Breaking Dawn]] contra [[Islamic Jihad Movement in Palestine|Palestinian Islamic Jihad]] in [[Gaza Strip]]: 20 militants and many civilians, with 1,100 rockets fired into Israels south and center, some misfiring and killing civilians in Gaza.<ref>{{Cite news|last=Boxerman|first=Dov Lieber and Aaron|date=2022-08-08|title=Israel Reopens Gaza Crossing as Cease-Fire Holds|language=en-US|work=Wall Street Journal|url=https://www.wsj.com/articles/israel-reopens-gaza-crossing-as-cease-fire-holds-11659972963|access-date=2022-08-11|issn=0099-9660}}</ref> == 5784 - 2024 in context == === Iranian democracy movements === [[Draft:Iranian democracy movements]]. [[w:Ayatollah_Ali_Khamenei|Ayatollah Ali Khamenei]] explains, ""It’s neither logical nor acceptable to the public opinion for an Iranian army to go and fight in place of the Syrian army. No, it’s the duty of the army of that country itself to fight. What our forces could do and did was to provide advisory support." He added: "The army of a nation must do the main fighting. The Basiji, voluntary forces from elsewhere can only fight alongside the army of that country. If the local army shows weakness, the Basij cannot do anything. Unfortunately, this is what happened in Syria." The young people of Syria will liberate the territories occupied by the Zionist entity," but he also admitted that ""[The] Zionist regime has bombed over 300 locations in Syria. In addition, it has occupied Syrian territories."<ref>{{Cite web|url=https://www.israelnationalnews.com/news/400666|title=Iranian Supreme Leader: Not logical for Iranian army to fight in place of the Syrian army|website=www.israelnationalnews.com|publisher=Arutz Sheva|access-date=2024-12-13}}</ref> === The [[w:Fall_of_the_Assad_regime#Israeli_invasion|2024 Fall of al-Assad Regime]] === '''The 1974 [[w:Agreement_on_Disengagement_between_Israel_and_Syria|Agreement on Disengagement between Israel and Syria]]''' ({{langx|he|הסכם הפרדת הכוחות בין ישראל לסוריה}}, {{langx|ar|اتفاقية فك الاشتباك}}) was an agreement between [[w:Israel|Israel]] and [[w:Syria|Syria]] that was signed on May 31, 1974,<ref name="Osmańczyk20032">{{cite book|url=https://books.google.com/books?id=aDwDmuOEheIC&pg=PA2263|title=Encyclopedia of the United Nations and International Agreements: A to F|author=Edmund Jan Osmańczyk|publisher=Taylor & Francis|year=2003|isbn=978-0-415-93921-8|pages=2263–}}</ref> which officially ended the [[w:Yom_Kippur_War|Yom Kippur War]] and the subsequent attrition period on the Syrian front.<ref name="United Nations Peacekeeping 19742">{{cite web|url=https://peacekeeping.un.org/en/mission/undof|title=UNDOF|date=1974-05-31|website=United Nations Peacekeeping|access-date=2018-02-20}}</ref> Following the [[w:Fall_of_the_Assad_regime|fall of the Assad regime]], Israel has considered the agreement null and void, leading to the [[w:2024_Israeli_invasion_of_Syria|2024 Israeli invasion of Syria]].<ref>{{Cite web|url=https://www.bbc.com/news/articles/c77jrrxxn07o|title=Israel seizes Golan buffer zone after Syrian troops leave posts|website=www.bbc.com|language=en-GB|access-date=2024-12-08}}</ref> WSJ: "An Israeli offensive aimed at rolling back Iranian influence pummeled Iran’s network of militia allies in the region, devastating Hamas in Gaza, Hezbollah in Lebanon and striking Iran’s shadow military network in Syria", which helped "opened the door for Hayat Tahrir al-Sham". "Iran told Assad help for his regime would be limited, if it came at all. Iranian officials blamed Assad for not preparing for the rebel assault and said they weren’t able to send military reinforcements because of Israel, according to Syrian officials. An Iranian plane headed toward Syria had to make a U-turn because of the threat of Israeli airstrikes, the officials said." "Rather than lend aid, Iran ordered its Islamic Revolutionary Guard Corps and affiliated militias to stay out of the fight, Syrian officials said. Iran then coordinated a safe exit for its personnel and cut a deal for its fighters to peacefully hand territory over to rebels."<ref>{{Cite news|url=http://www.wsj.com/world/middle-east/how-syria-rebels-ousted-assad-1b09d28d?page=1|title=The 11-Day Blitz by Syrian Rebels That Ended 50 Years of Assad Rule|date=9 December 2024|work=The Wall Street Journal}}</ref> [[w:he:שיחה:נפילת_משטר_אל-אסד_בסוריה#השפעת_כוחות_ישראל_בהתנגד_אירן]] [[w:he:שיחה:נפילת_משטר_אל-אסד_בסוריה#נשיא_אה"ב_ג'ו_ביידן_תבע_אשראי]] 8 December: [[s:Remarks_by_President_Biden_on_the_Latest_Developments_in_Syria]]. But, {{Cite news|url=https://www.wsj.com/opinion/joe-biden-syria-bashar-al-assad-white-house-israel-jake-sullivan-40f7db71?page=1|title=Biden Takes an Undeserved Syria Victory Lap|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}} === Lebanon: 2005 Cedar Revolution to the degradation of Hezbollah === 2005 [[w:Cedar_Revolution|Cedar Revolution]] ==Israel&mdash;Palestine== * [[Maccabean Revolt]] against the [[Seleucid Empire]] was fought by the [[Maccabees]] in [[Judea]] in 167-164 BCE, giving us [[Hannukah]]. *[[Arab-Israeli conflict]] 1948-1973 * [[Israeli-Palestinian conflict]] ongoing {{EGY}}<br> {{ISR}}<br />&nbsp;'''∟''' ''{{flagdeco|Israel}} [[Northern District (Israel)|Northern District]]''{{efn|[[Golan Heights]] was annexed to Israel's [[Northern District (Israel)|Northern District]] via [[Golan Heights Law]] but Israeli sovereignty over Golan is [[United Nations Security Council Resolution 497|not recognised internationally]].}}<br />&nbsp;'''∟''' ''{{flagicon image|Flag of the Israel Defense Forces.svg}} [[Judea and Samaria Area]]''<br />&nbsp;'''∟''' ''{{flagicon image|Flag of Jerusalem.svg}} [[East Jerusalem]]''{{efn|East Jerusalem was annexed to Israel via [[Jerusalem Law]] but Israeli sovereignty over Jerusalem is [[Status of Jerusalem|disputed]].}}<br /> {{JOR}}<br> {{LBN}}<br> {{flag|State of Palestine}}<br>{{small|([[Palestinian National Authority]])}}<br />&nbsp;'''∟''' {{flag|Gaza Strip}}{{efn|[[Governance of the Gaza Strip|Governed]] by [[Hamas]] [[Fatah–Hamas conflict|since 2007]].}}<br />&nbsp;'''∟''' {{flag|West Bank}}<br /> {{SYR}} * [[State of Palestine]], a ''de jure'' sovereign state in Western Asia * [[Palestine (region)]], a geographic region in Western Asia * [[Palestinian territories]], territories occupied by Israel since 1967, namely the West Bank (including East Jerusalem) and the Gaza Strip * [[Palestinian enclaves]], the areas designated for Palestinians under a variety of US and Israeli-led proposals * [[Mandatory Palestine]] (1920–1948), a geopolitical entity under British administration * [[Timeline of the name "Palestine"]] lists other historic uses == Palestine Etymological considerations == The [[Philistines]] or [[Peleset]] were one of the mediterranean [[Sea Peoples|Sea-peoples]] of probably Aegean or Greek origin, from [[Mycenaean Greece|Mycenaean civilization]]{{Cn|date=August 2022}}, who inhabited five major cities in Gaza coast in ancient times, and were thought by the Hebrews to have originated in [[Caphtor]] - Crete / Minoa. Per [[Martin Noth]], the name likely comes from the Aramaic word for Philistine. Noth also described a strong similarity between the word Palestine and the Greek word "palaistês" (wrestler/rival/adversary), which has the same meaning as the word "Israel."{{sfn|Noth|1939}} This was expanded by David Jacobson to theorize the name being a [[portmanteau]] of the word for Philistines with a direct translation of the word Israel into [[Greek language|Greek]] (in concordance with the Greek penchant for punning on place names.{{sfn|Jacobson|1999|p={{pn|date=February 2021}}|ps=: "In the earliest Classical literature references to Palestine generally applied to the Land of Israel in the wider sense. A reappraisal of this question has given rise to the proposition that the name Palestine, in its Greek form Palaistine, was both a transliteration of a word used to describe the land of the Philistines and, at the same time, a literal translation of the name Israel. This dual interpretation reconciles apparent contradictions in early definitions of the name Palaistine and is compatible with the Greeks' penchant for punning, especially on place names."}}<ref>{{cite book |last=Beloe |first=W. |title=Herodotus, Vol.II |location=London |year=1821 |page=269 |quote=It should be remembered that Syria is always regarded by Herodotus as synonymous with [[Assyria]]. What the Greeks called Palestine the Arabs call Falastin, which is the Philistines of Scripture. |url=https://books.google.com/books?id=SyYIAAAAQAAJ&pg=PA269}} (tr. from Greek, with notes)</ref> == References == <references group="lower-alpha" /> {{reflist}} ==Bibliography== * {{cite book|last=Louvish|first=Misha|title=The Challenge Of Israel|publisher=Israel Universities Press|location=Jerusalem|date=1968|ASIN=B000OKO5U2}} * {{cite book |last=Montefiore |first=Simon Sebag |title=Jerusalem: The Biography |title-link=Jerusalem: The Biography |date=2011 |publisher=Vintage Books, Random House |ISBN=978-0-307-28050-3 |author-link=Simon Sebag Montefiore}} {{refend}} s4zzvo368v2qiviyhpezcxkokb9ftf4 2691750 2691749 2024-12-13T07:43:54Z Jaredscribe 2906761 /* Iranian democracy movements */ 2691750 wikitext text/x-wiki This is reading list and study notes. resume is on [[User:Jaredscribe#Zionism, Palestine, Israel]] The [[Arab%E2%80%93Israeli alliance against Iran]] emerged in 2017. The [[February 2019 Warsaw Conference]] prepared the way for the [[Abraham Accords]], the [[Israel%E2%80%93United Arab Emirates normalization agreement]], and the {{slink|Arab%E2%80%93Israeli conflict#Israeli normalization with Gulf states and Sudan}} [[Negev Summit]] Recent phase of the [[Arab League and the Arab%E2%80%93Israeli conflict]] * 5 Iyar [[Independence Day (Israel)]], 15 May [[Nakba Day]] * 30 Nov [[Day to Mark the Departure and Expulsion of Jews from the Arab Countries and Iran]] * Libyan [[Day_of_Revenge]]/Friendship? [[Proto-Zionism]] == [[Mandatory Palestine]] == {{blockquote| text=We have watched the rise and fall of this Palestine problem. In 1917 Jewish hopes were raised in all parts of the world. It was thought that at long last here was the Jewish Magna Carta. By 1921 Transjordan was lopped off; in 1922 free immigration became immigration on the basis of absorptive capacity; in 1933 land sales were restricted; in 1937, partition was accepted by the Government; and in 1938 partition was rejected by the Government; and in 1939 we see the funeral of the mandate."| author=[[Tom Williams, Baron Williams of Barnburgh|Tom Williams]] was member of Churchhill's [[Coalition Government 1940-1945]], and later of the government of [[Clement Attlee]] }} [[Haganah|Hagana]] [[Jewish Supernumerary Police|Supernumerary police]] [[Orde Wingate]] [[Havlagah|Havlaga Policy of Restraint]] [[Irgun]] [[Lehi (militant group)|Lehi]] Stern Dr. [[Chaim Weizmann]] [[Peel Commission]] [[Woodhead Commission]] 17 May [[White Paper of 1939|1939 White Paper]] 21st [[World Zionist Congress|Zionist Congress]] 28 Feb 1940 British gov't issued the land transfer regulations envisioned in the White Paper: Palestine divided into Zone A (only to Palestinian Arab), Zone B (Gov't discretion), Zone C (no restrictions). [[Jewish National Fund]]. == Israeli Independence and Arab Catastrophe == [[London Conference of 1946–47]] boycotted except by Arab League states, who argue against partition. [[United Nations Special Committee on Palestine|UNSCOP]] majority proposal 8/11, Zionist side accepts, Arab side rejects. General Assembly appoints [[Ad Hoc Committee on the Palestinian Question]] which .. 29 Nov 1947 [[United Nations Partition Plan for Palestine]], Resolution 181. Accepted by the [[Jewish Agency for Israel|Jewish Agency]], rejected by Arab leaders. [[1947–1948 civil war in Mandatory Palestine]] breaks out. [[1947–1949 Palestine war]] duplicate? [[SS Exodus|SS Exodus 1947]] [[Arab Higher Committee]]: [[Grand Mufti of Jerusalem]] and president of [[Supreme Muslim Council]] [[Amin al-Husayni]], [[Raghib al-Nashashibi]] [[Plan Dalet]] March 1948 arms smuggling [[Operation Balak]] [[Va'ad Leumi]], [[Jewish Agency for Israel|Jewish Agency]], Haganah Command, prominent figures from trade unions, religion, literature and arts, representatives of [[Yishuv|the Yishuv]], at the [[Tel Aviv Museum|Tel Aviv Museum hall]] 5 Iyar / 14 May [[David Ben-Gurion]] proclaims [[Israeli Declaration of Independence]], [[s:Declaration of Independence (Israel)]] [[Bevingrad]] British general [[Alan Cunningham]] Levett Machal [[Sar-El|Sar-el]] [[1948 Arab–Israeli War]] == Divided 1951-1967 == [[King Abdullah of Jordan|King Abdullah]] crowned [[king of Jerusalem]] by the Coptic bishop. He had himself declared king of Palestine in Jericho, and renamed the realm "[[Kingdom of Jordan|United Kingdom of Jordan]]".{{Sfn|Montefiore|2011|p=504}} Appointed [[Raghib al-Nashashibi]] governer of West Bank and Custodian of the two harams (Jerusalem and Hebron) Appointed Sheikh [[Husam al-Jarallah]] as mufti. [[Farouk of Egypt|King Farouk]] overthrown by [[Free Officers Movement (Egypt)|Free Officers of Egypt]], led by general [[Mohamed Naguib|Muhammad Neguib]] and colonel [[Gamal Abdel Nasser|Gamel Abdel Nasser]]. == Six Days and Jerusalem == , [[United Arab Republic]] [[Straits of Tiran]] blockade, [[Israeli passage through the Suez Canal and Straits of Tiran]]. [[Six day war]] [[Egypt–Israel peace treaty]], [[Israel–Jordan peace treaty]], [[Israel–Saudi Arabia relations#Straits of Tiran]] ==5782&mdash;2022== Blinken and Lapid would head to Sde Boker for the [[w:Negev_Summit|Negev Summit]], along with their Bahraini, Egyptian, Emirati and Moroccan counterparts. Blinken called the Houthis’ attack on Iran “acts of terrorism enabled by Iran.” Blinken also pointed out that Iran continues to engage in aggressive actions throughout the Middle East and beyond, directly and through proxies, and spoke out against “mounting terrorist attacks by the Houthis on civilians and civilian infrastructure in the UAE and Saudi Arabia.” “The US will continue to stand up to Iran when it threatens us or our allies and partners,” Blinken said. The US believes “the [[w:JCPOA|JCPOA]] is the best way to put Iran back in the box,” Blinken said, referring to the 2015 nuclear deal to which the US seeks to return, and Israel opposes. Regardless of whether that effort succeeds, he said, “our commitment to the core principle that Iran can never acquire a nuclear weapon is unwavering.” “One way or another, we will continue to cooperate closely,” he stated. “Iran is not an Israeli problem,” Lapid said. “The whole world cannot allow there to be a nuclear Iran and cannot allow the Iranian Revolutionary Guard Corps to continue to spread terror across the globe.” Lapid: “Israel will do everything it thinks is right in order to stop the Iran nuclear program. Everything.” Lapid: “That is not a theoretical threat for us,” he added. “The Iranians want to destroy Israel. They won’t succeed. We won’t let them.” Blinken, US secretary of state: "[the Negev Summit ] would have been unthinkable just a few years ago,” and said the US is committed to expanding cooperation through the Abraham Accords. Blinken: "[will] affirm America’s ironclad commitment to Israel’s security.” Biden signed the omnibus funding bill that included $1 billion for the Iron Dome missile defense system. [https://www.jpost.com/israel-news/politics-and-diplomacy/article-702431 JPost Negev Summit] [https://www.nytimes.com/2022/03/25/world/middleeast/israel-uae-bahrain-morocco-arab-summit.html NYTimes Negev Summit] 5 August 2022 [[Operation Breaking Dawn]] contra [[Islamic Jihad Movement in Palestine|Palestinian Islamic Jihad]] in [[Gaza Strip]]: 20 militants and many civilians, with 1,100 rockets fired into Israels south and center, some misfiring and killing civilians in Gaza.<ref>{{Cite news|last=Boxerman|first=Dov Lieber and Aaron|date=2022-08-08|title=Israel Reopens Gaza Crossing as Cease-Fire Holds|language=en-US|work=Wall Street Journal|url=https://www.wsj.com/articles/israel-reopens-gaza-crossing-as-cease-fire-holds-11659972963|access-date=2022-08-11|issn=0099-9660}}</ref> == 5784 - 2024 in context == === Iranian democracy movements === [[Draft:Iranian democracy movements]]. Israeli Prime minister Benjamin Netanyahu addressed the Iranina people again, in another English speech with Farsi subtitles, saying, "As we see history unfold before our very eyes, I can only imagine what you're feeling right now. Your oppressors spent over 30 billion dollars supporting Assad in Syria. Today, after only 11 days of fighting, his regime collapsed into the dust," Netanyahu opened. "You're oppressors spent billions supporting Hamas in Gaza. Today, their regime lies in ruins. Your oppressors spent over 20 billion dollars supporting Hezbollah in Lebanon. In a matter of weeks, most of Hezbollah's leaders, its rockets, and thousands of its terrorists went up in smoke. It's the money you're oppressors stole from you that truly went up in smoke" ... "You know what this regime is truly terrified of? It's terrified of you, the people of Iran. And one day, I know that. One day this will change. One day Iran will be free," he predicted. "Women, Life, Freedom Zan, Zendegi, Azadi. That is the future of Iran. That is the future of peace. And I have no doubt that we will realize that future together – a lot sooner than people think. I know and I believe we will transform the Middle East into a beacon of prosperity, progress, and peace," the Prime Minister concluded.<ref>{{Cite web|url=https://www.israelnationalnews.com/news/400663|title=Netanyahu to Iranians: 'The billions your oppressors stole from you went up in smoke'|website=www.israelnationalnews.com|publisher=Arutz Sheva|access-date=2024-12-13}}</ref> [[w:Ayatollah_Ali_Khamenei|Ayatollah Ali Khamenei]] explains, ""It’s neither logical nor acceptable to the public opinion for an Iranian army to go and fight in place of the Syrian army. No, it’s the duty of the army of that country itself to fight. What our forces could do and did was to provide advisory support." He added: "The army of a nation must do the main fighting. The Basiji, voluntary forces from elsewhere can only fight alongside the army of that country. If the local army shows weakness, the Basij cannot do anything. Unfortunately, this is what happened in Syria." The young people of Syria will liberate the territories occupied by the Zionist entity," but he also admitted that ""[The] Zionist regime has bombed over 300 locations in Syria. In addition, it has occupied Syrian territories."<ref>{{Cite web|url=https://www.israelnationalnews.com/news/400666|title=Iranian Supreme Leader: Not logical for Iranian army to fight in place of the Syrian army|website=www.israelnationalnews.com|publisher=Arutz Sheva|access-date=2024-12-13}}</ref> === The [[w:Fall_of_the_Assad_regime#Israeli_invasion|2024 Fall of al-Assad Regime]] === '''The 1974 [[w:Agreement_on_Disengagement_between_Israel_and_Syria|Agreement on Disengagement between Israel and Syria]]''' ({{langx|he|הסכם הפרדת הכוחות בין ישראל לסוריה}}, {{langx|ar|اتفاقية فك الاشتباك}}) was an agreement between [[w:Israel|Israel]] and [[w:Syria|Syria]] that was signed on May 31, 1974,<ref name="Osmańczyk20032">{{cite book|url=https://books.google.com/books?id=aDwDmuOEheIC&pg=PA2263|title=Encyclopedia of the United Nations and International Agreements: A to F|author=Edmund Jan Osmańczyk|publisher=Taylor & Francis|year=2003|isbn=978-0-415-93921-8|pages=2263–}}</ref> which officially ended the [[w:Yom_Kippur_War|Yom Kippur War]] and the subsequent attrition period on the Syrian front.<ref name="United Nations Peacekeeping 19742">{{cite web|url=https://peacekeeping.un.org/en/mission/undof|title=UNDOF|date=1974-05-31|website=United Nations Peacekeeping|access-date=2018-02-20}}</ref> Following the [[w:Fall_of_the_Assad_regime|fall of the Assad regime]], Israel has considered the agreement null and void, leading to the [[w:2024_Israeli_invasion_of_Syria|2024 Israeli invasion of Syria]].<ref>{{Cite web|url=https://www.bbc.com/news/articles/c77jrrxxn07o|title=Israel seizes Golan buffer zone after Syrian troops leave posts|website=www.bbc.com|language=en-GB|access-date=2024-12-08}}</ref> WSJ: "An Israeli offensive aimed at rolling back Iranian influence pummeled Iran’s network of militia allies in the region, devastating Hamas in Gaza, Hezbollah in Lebanon and striking Iran’s shadow military network in Syria", which helped "opened the door for Hayat Tahrir al-Sham". "Iran told Assad help for his regime would be limited, if it came at all. Iranian officials blamed Assad for not preparing for the rebel assault and said they weren’t able to send military reinforcements because of Israel, according to Syrian officials. An Iranian plane headed toward Syria had to make a U-turn because of the threat of Israeli airstrikes, the officials said." "Rather than lend aid, Iran ordered its Islamic Revolutionary Guard Corps and affiliated militias to stay out of the fight, Syrian officials said. Iran then coordinated a safe exit for its personnel and cut a deal for its fighters to peacefully hand territory over to rebels."<ref>{{Cite news|url=http://www.wsj.com/world/middle-east/how-syria-rebels-ousted-assad-1b09d28d?page=1|title=The 11-Day Blitz by Syrian Rebels That Ended 50 Years of Assad Rule|date=9 December 2024|work=The Wall Street Journal}}</ref> [[w:he:שיחה:נפילת_משטר_אל-אסד_בסוריה#השפעת_כוחות_ישראל_בהתנגד_אירן]] [[w:he:שיחה:נפילת_משטר_אל-אסד_בסוריה#נשיא_אה"ב_ג'ו_ביידן_תבע_אשראי]] 8 December: [[s:Remarks_by_President_Biden_on_the_Latest_Developments_in_Syria]]. But, {{Cite news|url=https://www.wsj.com/opinion/joe-biden-syria-bashar-al-assad-white-house-israel-jake-sullivan-40f7db71?page=1|title=Biden Takes an Undeserved Syria Victory Lap|last=Editorial Board|date=10 December 2024|work=The Wall Street Journal}} === Lebanon: 2005 Cedar Revolution to the degradation of Hezbollah === 2005 [[w:Cedar_Revolution|Cedar Revolution]] ==Israel&mdash;Palestine== * [[Maccabean Revolt]] against the [[Seleucid Empire]] was fought by the [[Maccabees]] in [[Judea]] in 167-164 BCE, giving us [[Hannukah]]. *[[Arab-Israeli conflict]] 1948-1973 * [[Israeli-Palestinian conflict]] ongoing {{EGY}}<br> {{ISR}}<br />&nbsp;'''∟''' ''{{flagdeco|Israel}} [[Northern District (Israel)|Northern District]]''{{efn|[[Golan Heights]] was annexed to Israel's [[Northern District (Israel)|Northern District]] via [[Golan Heights Law]] but Israeli sovereignty over Golan is [[United Nations Security Council Resolution 497|not recognised internationally]].}}<br />&nbsp;'''∟''' ''{{flagicon image|Flag of the Israel Defense Forces.svg}} [[Judea and Samaria Area]]''<br />&nbsp;'''∟''' ''{{flagicon image|Flag of Jerusalem.svg}} [[East Jerusalem]]''{{efn|East Jerusalem was annexed to Israel via [[Jerusalem Law]] but Israeli sovereignty over Jerusalem is [[Status of Jerusalem|disputed]].}}<br /> {{JOR}}<br> {{LBN}}<br> {{flag|State of Palestine}}<br>{{small|([[Palestinian National Authority]])}}<br />&nbsp;'''∟''' {{flag|Gaza Strip}}{{efn|[[Governance of the Gaza Strip|Governed]] by [[Hamas]] [[Fatah–Hamas conflict|since 2007]].}}<br />&nbsp;'''∟''' {{flag|West Bank}}<br /> {{SYR}} * [[State of Palestine]], a ''de jure'' sovereign state in Western Asia * [[Palestine (region)]], a geographic region in Western Asia * [[Palestinian territories]], territories occupied by Israel since 1967, namely the West Bank (including East Jerusalem) and the Gaza Strip * [[Palestinian enclaves]], the areas designated for Palestinians under a variety of US and Israeli-led proposals * [[Mandatory Palestine]] (1920–1948), a geopolitical entity under British administration * [[Timeline of the name "Palestine"]] lists other historic uses == Palestine Etymological considerations == The [[Philistines]] or [[Peleset]] were one of the mediterranean [[Sea Peoples|Sea-peoples]] of probably Aegean or Greek origin, from [[Mycenaean Greece|Mycenaean civilization]]{{Cn|date=August 2022}}, who inhabited five major cities in Gaza coast in ancient times, and were thought by the Hebrews to have originated in [[Caphtor]] - Crete / Minoa. Per [[Martin Noth]], the name likely comes from the Aramaic word for Philistine. Noth also described a strong similarity between the word Palestine and the Greek word "palaistês" (wrestler/rival/adversary), which has the same meaning as the word "Israel."{{sfn|Noth|1939}} This was expanded by David Jacobson to theorize the name being a [[portmanteau]] of the word for Philistines with a direct translation of the word Israel into [[Greek language|Greek]] (in concordance with the Greek penchant for punning on place names.{{sfn|Jacobson|1999|p={{pn|date=February 2021}}|ps=: "In the earliest Classical literature references to Palestine generally applied to the Land of Israel in the wider sense. A reappraisal of this question has given rise to the proposition that the name Palestine, in its Greek form Palaistine, was both a transliteration of a word used to describe the land of the Philistines and, at the same time, a literal translation of the name Israel. This dual interpretation reconciles apparent contradictions in early definitions of the name Palaistine and is compatible with the Greeks' penchant for punning, especially on place names."}}<ref>{{cite book |last=Beloe |first=W. |title=Herodotus, Vol.II |location=London |year=1821 |page=269 |quote=It should be remembered that Syria is always regarded by Herodotus as synonymous with [[Assyria]]. What the Greeks called Palestine the Arabs call Falastin, which is the Philistines of Scripture. |url=https://books.google.com/books?id=SyYIAAAAQAAJ&pg=PA269}} (tr. from Greek, with notes)</ref> == References == <references group="lower-alpha" /> {{reflist}} ==Bibliography== * {{cite book|last=Louvish|first=Misha|title=The Challenge Of Israel|publisher=Israel Universities Press|location=Jerusalem|date=1968|ASIN=B000OKO5U2}} * {{cite book |last=Montefiore |first=Simon Sebag |title=Jerusalem: The Biography |title-link=Jerusalem: The Biography |date=2011 |publisher=Vintage Books, Random House |ISBN=978-0-307-28050-3 |author-link=Simon Sebag Montefiore}} {{refend}} 0gai6znzicxyz05yuxojmslen788th3 0 317149 2691622 2691521 2024-12-12T13:32:58Z Eshaa2024 2993595 /* See also */ 2691622 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[v:de:Kurs:Funktionentheorie]] * [[v:de:Kurs:Funktionentheorie/Quiz]] == Sources == 4p13n7uqtdiobzu5l8b3kpki6hzms4s 2691744 2691622 2024-12-13T07:29:07Z Bert Niehaus 2387134 Bert Niehaus moved page [[Holomorphism/Criteria]] to [[Holomorphic function/Criteria]]: move subpage to appropriate parent page "Holomorphic function" 2691622 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[v:de:Kurs:Funktionentheorie]] * [[v:de:Kurs:Funktionentheorie/Quiz]] == Sources == 4p13n7uqtdiobzu5l8b3kpki6hzms4s 2691746 2691744 2024-12-13T07:29:54Z Bert Niehaus 2387134 /* See also */ 2691746 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[Complex Analysis]] * [[Complex Analysis/Quiz]] == Sources == bkcokh428zp372rp2g0111wepbao2n2 2691764 2691746 2024-12-13T08:24:07Z Bert Niehaus 2387134 /* Sources */ 2691764 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[Complex Analysis]] * [[Complex Analysis/Quiz]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Holomorphic%20function Holomorphic function]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function/Criteria https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] 5nrufhc8si3o5wqcqqhjwoq15frbvjj 2691769 2691764 2024-12-13T08:49:54Z Bert Niehaus 2387134 /* Page Information */ 2691769 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[Complex Analysis]] * [[Complex Analysis/Quiz]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Holomorphic%20function Holomorphic function]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function/Criteria https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphie/Kriterien Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Holomorphie/Kriterien|Holomorphie/Kriterien]] * URL: https://de.wikiversity.org/wiki/Holomorphie/Kriterien * Date: 12/13/2024 9:48 <span type="translate" src="Holomorphie/Kriterien" srclang="de" date="12/13/2024" time="9:48" status="inprogress"></span> <noinclude>[[de:Holomorphie/Kriterien]]</noinclude> <!-- <noinclude>[[en:Holomorphic function/Criteria]]</noinclude> --> [[Category:Wiki2Reveal]] hz0s6luy9vky7c019msqfz5yoko6qih 2691770 2691769 2024-12-13T08:51:02Z Bert Niehaus 2387134 , 10 December 2024/* Translation and Version Control */ 2691770 wikitext text/x-wiki == Introduction == Holomorphy of a function <math>f\colon U \to \mathbb{C}</math> at a point <math>z_0 \in U</math> is a neighborhood property of <math>z_0</math>. There are numerous criteria in complex analysis that can be used to verify holomorphy. Let <math>U \subseteq \mathbb{C}</math> be a domain as a subset of the complex plane and <math>z_0 \in U</math> a point in this subset. === Animation - Visualization of the Mapping === The animation shows the function <math>f(z) = \frac{1}{z}</math>. In the animation, <math>z</math> is shown in blue, and the corresponding image point <math>f(z)</math> is shown in red. The point <math>z</math> and <math>f(z)</math> are represented in <math>\mathbb{C} \widetilde{=} \mathbb{R}^2</math>. The <math>y</math>-axis represents the imaginary part of the complex numbers <math>z</math> and <math>f(z)</math>. The blue point <math>z</math> moves along the path <math>\gamma(t) := t \cdot (\cos(t) + i \cdot \sin(t))</math> [[File:Mapping f z equal 1 over z.gif|400px|centered|Animation]] === Complex Differentiability === A function <math>f\colon U \to \mathbb{C}</math> is called '''complex differentiable''' at the point <math>z_0</math> if the limit <math>\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}</math> exists with <math>h \in \mathbb{C}</math>. This is denoted as <math>f'(z_0)</math>. === Holomorphy === A function <math>f\colon U \to \mathbb{C}</math> is called '''holomorphic at the point <math>z_0</math>''' if there exists a neighborhood <math>U_0 \subseteq U</math> of <math>z_0</math> such that <math>f</math> is complex differentiable in <math>U_0</math>. If <math>f</math> is holomorphic on all of <math>U</math>, it is simply called holomorphic. If additionally <math>U = \mathbb{C}</math>, <math>f</math> is called an '''entire function'''. == Holomorphy Criteria == Let <math>f: U \to \mathbb{C}</math> be a function where <math>U \subseteq \mathbb{C}</math> is a domain, then the following properties of the complex-valued function <math>f</math> are equivalent: === (HK1) Once Complex Differentiable === The function <math>f</math> is once complex differentiable on <math>U</math>. === (HK2) Arbitrarily Often Complex Differentiable === The function <math>f</math> is arbitrarily often complex differentiable on <math>U</math>. === (HK3) Cauchy-Riemann Differential Equations === The real and imaginary parts satisfy the [[w:en:Cauchy-Riemann equations|Cauchy-Riemann equations]] and are at least once continuously real-differentiable on <math>U</math>. === (HK4) Locally Expansible in Power Series === The function can be locally expanded in a complex [[w:Power series|power series]] on <math>U</math>. === (HK5) Path Integrals 0 === The function <math>f</math> is continuous, and the [[w:Path integral|path integral]] of the function over any closed [[w:Homotopy|contractible]] path vanishes (i.e., the winding number of the path integral for all points outside of <math>U</math> is 0). === (HK6) Cauchy Integral Formula === The function values inside a [[w:en:circular disk|circular disk]] can be determined from the function values on the boundary using the [[w:en:Cauchy Integralformel|Cauchy integral formula]]. === (HK7) Cauchy-Riemann Operator === <math>f</math> is real differentiable, and <math>\frac{\partial f}{\partial \bar{z}} = 0</math>, where <math>\frac{\partial}{\partial \bar{z}}</math> is the [[w:en:Cauchy-Riemann operator|Cauchy-Riemann operator]] defined by <math>\frac{\partial}{\partial \bar{z}} := \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)</math>. == Exercises == Let <math>a, z_0 \in \mathbb{C}</math> be chosen arbitrarily, and assume that <math>a \neq z_0</math>. Now, develop the function <math>f(z) := \frac{1}{z - a}</math> for <math>z \in \mathbb{C} \setminus {a}</math> in a power series around <math>z_0 \in \mathbb{C}</math> and show that the following holds: <math display="block">f(z) = \frac{1}{z - a} = \sum_{n=0}^{\infty} -\frac{1}{(a - z_0)^{n + 1}} \cdot (z - z_0)^n</math> Calculate the radius of convergence of the power series! Explain why the [[w:en:Radius of convergence|radius of convergence]] depends on <math>a, z_0 \in \mathbb{C}</math> in this way and cannot be larger! It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function <math>g(x) := x \cdot |x|</math> defined on all of <math>\mathbb{R}</math>. Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2! == See also == * [[Complex Analysis]] * [[Complex Analysis/Quiz]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Holomorphic%20function Holomorphic function]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Holomorphic%20function/Criteria https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria] --> * [https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic%20function/Criteria * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Holomorphic%20function&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Holomorphic%20function Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Holomorphie/Kriterien Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Holomorphie/Kriterien|Holomorphie/Kriterien]] * URL: https://de.wikiversity.org/wiki/Holomorphie/Kriterien * Date: 12/10/2024 9:56pm <span type="translate" src="Holomorphie/Kriterien" srclang="de" date="12/09/2024" time="21:56" status="inprogress"></span> <noinclude>[[de:Holomorphie/Kriterien]]</noinclude> <!-- <noinclude>[[en:Holomorphic function/Criteria]]</noinclude> --> [[Category:Wiki2Reveal]] 2ivpdq49p6xzpivm1ayjsmfncq698eg 0 317156 2691620 2691375 2024-12-12T13:23:29Z Eshaa2024 2993595 /* Page Information */ 2691620 wikitext text/x-wiki == Introduction == Inequalities are an essential tool for proving central statements in function theory. Since <math>\mathbb{C}</math> does not have a [[w:en:Order_relation#Total_order|complete/total order]], one must rely on the magnitude of functions for estimations. == Inequality for Piecewise Continuous Functions == Let <math>f:[a,b] \rightarrow \mathbb{C}</math> be piecewise continuous. Then, the following holds:<ref>Funktionentheorie, Fischer, W., Lieb, W. (1988) Vieweg, p. 37</ref> :<math>\left| \int_{a}^{b} f(t), dt \right| \leq \int_{a}^{b} | f(t) | , dt </math> == Inequality for Estimation Over Integration Paths == Let <math>\gamma:[a,b] \rightarrow \mathbb{C}</math> be an [[w:Course:Functiontheory/Integration path|Integration path]] and <math>f:U \rightarrow \mathbb{C}</math> a continuous function on the trace of <math>\gamma</math> (<math>Trace(\gamma):={\gamma(t) , : , t \in [a,b]} \subset U</math>). Then, the following holds: :<math>\left| \int_{\gamma} f(z), dz \right| \leq \max_{z \in Trace(\gamma)} | f(z) | \cdot L(\gamma) </math> Here, <math>L(\gamma)= \int_{a}^{b} |\gamma'(t)| , dt</math> is the length of the integral. == See also == *[[Complex Analysis/rectifiable curve|rectifiable curve]] == Literature == <references/> == Page Information == This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Ungleichungen&author=Kurs:Funktionalanalysis&language=de&audioslide=yes Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/_Kurs:Funktionalanalysis Course:Functional Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. The content of the page is based on the following material: ** [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen] The page was created as a document type [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. [[Category:Wiki2Reveal]] [[Category:Functional Analysis]] mo3wcy6o8um3e79hr85qm3tiyipvot2 2691621 2691620 2024-12-12T13:24:03Z Eshaa2024 2993595 /* Literature */ 2691621 wikitext text/x-wiki == Introduction == Inequalities are an essential tool for proving central statements in function theory. Since <math>\mathbb{C}</math> does not have a [[w:en:Order_relation#Total_order|complete/total order]], one must rely on the magnitude of functions for estimations. == Inequality for Piecewise Continuous Functions == Let <math>f:[a,b] \rightarrow \mathbb{C}</math> be piecewise continuous. Then, the following holds:<ref>Funktionentheorie, Fischer, W., Lieb, W. (1988) Vieweg, p. 37</ref> :<math>\left| \int_{a}^{b} f(t), dt \right| \leq \int_{a}^{b} | f(t) | , dt </math> == Inequality for Estimation Over Integration Paths == Let <math>\gamma:[a,b] \rightarrow \mathbb{C}</math> be an [[w:Course:Functiontheory/Integration path|Integration path]] and <math>f:U \rightarrow \mathbb{C}</math> a continuous function on the trace of <math>\gamma</math> (<math>Trace(\gamma):={\gamma(t) , : , t \in [a,b]} \subset U</math>). Then, the following holds: :<math>\left| \int_{\gamma} f(z), dz \right| \leq \max_{z \in Trace(\gamma)} | f(z) | \cdot L(\gamma) </math> Here, <math>L(\gamma)= \int_{a}^{b} |\gamma'(t)| , dt</math> is the length of the integral. == See also == *[[Complex Analysis/rectifiable curve|rectifiable curve]] == Literature == <references/> == Page Information == This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Ungleichungen&author=Kurs:Funktionalanalysis&language=de&audioslide=yes Wiki2Reveal slide set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/_Kurs:Funktionalanalysis Course:Functional Analysis]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. The content of the page is based on the following material: ** [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen] The page was created as a document type [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Ungleichungen See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. [[Category:Wiki2Reveal]] [[Category:Functional Analysis]] gz5yfbarwilvtcpw02enju01z24g1qa 0 317158 2691612 2691607 2024-12-12T12:54:54Z Bert Niehaus 2387134 /* Translation and Version Control */ 2691612 wikitext text/x-wiki Rectifiable curves are an important term from the theory of [[Path_Integral|Path Integrals]]. They are those curves that can occur as an integration area. ==Definition== Let <math display="inline"> \gamma\colon [a,b]\to \mathbb C </math> a continuous [[Complex Analysis/Path|curve]]. A curve is called ''rectifiable'' if its length exist <center><math display="inline"> \mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> Finally, <math display="inline"> \mathcal{L}(\gamma) </math> is called 'length' of <math display="inline"> \gamma </math>. ===Examples=== * If <math display="inline"> \gamma </math> is continuously differentiable, then <math display="inline"> \gamma </math> is rectifiable. If <math display="inline"> a \le t_0 < \ldots < t_n \le b </math>, then the mean value <math display="inline"> \tau_i \in (t_{i-1}, t_i) </math> is such that <center><math display="inline"> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| (t_{i} - t_{i-1}) </math></center>The right side of the above equation is a Riemannian sum for <math display="inline"> \int_a^b |\gamma'(t)|\,dt </math>, i.e. * More generally, parts <math display="inline"> C^1 </math> curves are always rectifiable, the above consideration is applied to the individual parts of the curve. * As an example of a non-recipable curve, see <math display="inline"> \gamma\colon[0,1]\to \mathbb C </math>, <center><math display="inline"> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center>In the near future <math display="inline"> \gamma </math> is steadily differentiable and at any interval <math display="inline"> [\epsilon,1] </math>. At these intervals, the length is <center> <math display="inline"> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> For <math display="inline"> \epsilon > 0 </math> this converges against <center><math display="inline"> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center>so <math display="inline"> \gamma </math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * [[wikiversity:Course:Complex_Analysis/rectifiable Curve|Complex_Analysis/rectifiable Curve]] https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude>[[en:Course:Complex_Analysis/rectifiable Curve]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Recognition Curve]]</noinclude> --> f0byoihni9gsa5jiladt14pokue5u4m 2691613 2691612 2024-12-12T12:57:46Z Bert Niehaus 2387134 /* Translation and Version Control */ 2691613 wikitext text/x-wiki Rectifiable curves are an important term from the theory of [[Path_Integral|Path Integrals]]. They are those curves that can occur as an integration area. ==Definition== Let <math display="inline"> \gamma\colon [a,b]\to \mathbb C </math> a continuous [[Complex Analysis/Path|curve]]. A curve is called ''rectifiable'' if its length exist <center><math display="inline"> \mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> Finally, <math display="inline"> \mathcal{L}(\gamma) </math> is called 'length' of <math display="inline"> \gamma </math>. ===Examples=== * If <math display="inline"> \gamma </math> is continuously differentiable, then <math display="inline"> \gamma </math> is rectifiable. If <math display="inline"> a \le t_0 < \ldots < t_n \le b </math>, then the mean value <math display="inline"> \tau_i \in (t_{i-1}, t_i) </math> is such that <center><math display="inline"> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| (t_{i} - t_{i-1}) </math></center>The right side of the above equation is a Riemannian sum for <math display="inline"> \int_a^b |\gamma'(t)|\,dt </math>, i.e. * More generally, parts <math display="inline"> C^1 </math> curves are always rectifiable, the above consideration is applied to the individual parts of the curve. * As an example of a non-recipable curve, see <math display="inline"> \gamma\colon[0,1]\to \mathbb C </math>, <center><math display="inline"> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center>In the near future <math display="inline"> \gamma </math> is steadily differentiable and at any interval <math display="inline"> [\epsilon,1] </math>. At these intervals, the length is <center> <math display="inline"> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> For <math display="inline"> \epsilon > 0 </math> this converges against <center><math display="inline"> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center>so <math display="inline"> \gamma </math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/rektifizierbare Kurve|rektifizierbare Kurve]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/rektifizierbare Kurve]]</noinclude> fcicr09kn6j7bskmp6upew6phzqcjl2 2691614 2691613 2024-12-12T12:58:01Z Bert Niehaus 2387134 /* Translation and Version Control */ 2691614 wikitext text/x-wiki Rectifiable curves are an important term from the theory of [[Path_Integral|Path Integrals]]. They are those curves that can occur as an integration area. ==Definition== Let <math display="inline"> \gamma\colon [a,b]\to \mathbb C </math> a continuous [[Complex Analysis/Path|curve]]. A curve is called ''rectifiable'' if its length exist <center><math display="inline"> \mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> Finally, <math display="inline"> \mathcal{L}(\gamma) </math> is called 'length' of <math display="inline"> \gamma </math>. ===Examples=== * If <math display="inline"> \gamma </math> is continuously differentiable, then <math display="inline"> \gamma </math> is rectifiable. If <math display="inline"> a \le t_0 < \ldots < t_n \le b </math>, then the mean value <math display="inline"> \tau_i \in (t_{i-1}, t_i) </math> is such that <center><math display="inline"> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| (t_{i} - t_{i-1}) </math></center>The right side of the above equation is a Riemannian sum for <math display="inline"> \int_a^b |\gamma'(t)|\,dt </math>, i.e. * More generally, parts <math display="inline"> C^1 </math> curves are always rectifiable, the above consideration is applied to the individual parts of the curve. * As an example of a non-recipable curve, see <math display="inline"> \gamma\colon[0,1]\to \mathbb C </math>, <center><math display="inline"> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center>In the near future <math display="inline"> \gamma </math> is steadily differentiable and at any interval <math display="inline"> [\epsilon,1] </math>. At these intervals, the length is <center> <math display="inline"> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> For <math display="inline"> \epsilon > 0 </math> this converges against <center><math display="inline"> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center>so <math display="inline"> \gamma </math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/rektifizierbare Kurve|rektifizierbare Kurve]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/rektifizierbare Kurve]]</noinclude> 5r0w99a5c5xgtk9hru9d54awh0qzvrp 2691671 2691614 2024-12-12T18:00:19Z Eshaa2024 2993595 2691671 wikitext text/x-wiki == Definition == Let <math>\gamma\colon [a,b]\to \mathbb C</math> be a continuous curve. It is called rectifiable if its length <center><math>\mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> is finite, and <math>\mathcal{L}(\gamma)</math> is called the length of <math>\gamma</math>. === Approximation of path length by polygonal chain === The following image shows how a polygonal chain <math>P</math> can be used to approximate the length of a curve <math>\gamma</math>. [[File:Laenge kurve rektifizierbarkeit.png|350px|rectifiable curve - approximation of length by polygonal chain - created with Geogebra on Linux]] === Estimation of length === The length of the polygonal chain <math>P_n</math> underestimates the actual length of a rectifiable curve <math>\gamma</math>, i.e. <math>\mathcal{L}(P_n) \leq \mathcal{L}(\gamma)</math>. In general, <math>\mathcal{L}(P_n) < \mathcal{L}(\gamma)</math>. By applying the triangle inequality, we get <math> < </math> if the path's trace is not a line. == Path length for differentiable paths == If <math>\gamma</math> is continuously differentiable, then <math>\gamma</math> is rectifiable. Let <math>a \le t_0 < \ldots < t_n \le b</math>, then there exists <math>\tau_i \in (t_{i-1}, t_i)</math> such that <center><math> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| \cdot (t_{i} - t_{i-1}) </math></center> === Riemann sum as length of polygonal chain === The right-hand side of the above equation for the polygonal chain is a Riemann sum for the integral <math>\int_a^b |\gamma{\,}'(t)|\,dt</math>. If we take the maximum of the interval widths <math> \max_{i\in\{1, \ldots , n\} } (t_{i}-t_{t-1}) </math> for <math>n</math> to infinity, the length of the polygonal chains <math>\mathcal{L}(P_n)</math> converges to the length of the path <math> \mathcal{L}(\gamma) </math> === Length for continuously differentiable paths === Let <math>\gamma : [a,b] \to \mathbb{C} </math> be a continuously differentiable path, then <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt </math></center> gives the length of the path <math>\gamma</math>. === Note - Length for continuously differentiable paths === Since <math>\gamma</math> is continuously differentiable, <math>\gamma{\,}'</math> is a continuous function. Since <math>[a,b]</math> is a compact interval, <math>\gamma{\,}'</math> takes a minimum and maximum. Therefore, <math>\gamma{\,}'</math> and <math>|\gamma{\,}'|</math> are bounded, and we have: <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt < \infty </math></center> === Piecewise continuously differentiable curves === In general, piecewise <math>C^1</math>-curves are always rectifiable, because we can apply the above considerations to the individual parts of the curve, which then additively give the length of the entire curve. In the further course of complex analysis, paths (e.g. over the triangle edge) are considered that only possess the property of continuous differentiability in a piecewise manner, for which we can then still calculate the length as the sum of the arc lengths. == Non-rectifiable curve == As an example of a non-rectifiable curve, consider <math>\gamma\colon[0,1]\to \mathbb C</math>, <center><math> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center> === Continuity - continuous differentiability === First, <math>\gamma</math> is continuous and, on each interval <math>[\epsilon,1]</math>, even continuously differentiable. On these intervals, the length is given by <center><math> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> === Calculation of improper integral === For <math>\epsilon > 0</math>, this converges to <center><math> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center> so <math>\gamma</math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/rektifizierbare Kurve|rektifizierbare Kurve]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/rektifizierbare Kurve]] </noinclude> 2k62zar951u9r1i1dnnqvrmo1dzcmg8 2691777 2691671 2024-12-13T11:48:38Z Eshaa2024 2993595 /* Path length for differentiable paths */ 2691777 wikitext text/x-wiki == Definition == Let <math>\gamma\colon [a,b]\to \mathbb C</math> be a continuous curve. It is called rectifiable if its length <center><math>\mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> is finite, and <math>\mathcal{L}(\gamma)</math> is called the length of <math>\gamma</math>. === Approximation of path length by polygonal chain === The following image shows how a polygonal chain <math>P</math> can be used to approximate the length of a curve <math>\gamma</math>. [[File:Laenge kurve rektifizierbarkeit.png|350px|rectifiable curve - approximation of length by polygonal chain - created with Geogebra on Linux]] === Estimation of length === The length of the polygonal chain <math>P_n</math> underestimates the actual length of a rectifiable curve <math>\gamma</math>, i.e. <math>\mathcal{L}(P_n) \leq \mathcal{L}(\gamma)</math>. In general, <math>\mathcal{L}(P_n) < \mathcal{L}(\gamma)</math>. By applying the triangle inequality, we get <math> < </math> if the path's trace is not a line. == Path length for differentiable paths == If <math>\gamma</math> is continuously differentiable, then <math>\gamma</math> is rectifiable. Let <math>a \le t_0 < \ldots < t_n \le b</math>, then there exists[[w:en:Meanvalue|Meanvalue]] <math>\tau_i \in (t_{i-1}, t_i)</math> such that <center><math> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| \cdot (t_{i} - t_{i-1}) </math></center> === Riemann sum as length of polygonal chain === The right-hand side of the above equation for the polygonal chain is a Riemann sum for the integral <math>\int_a^b |\gamma{\,}'(t)|\,dt</math>. If we take the maximum of the interval widths <math> \max_{i\in\{1, \ldots , n\} } (t_{i}-t_{t-1}) </math> for <math>n</math> to infinity, the length of the polygonal chains <math>\mathcal{L}(P_n)</math> converges to the length of the path <math> \mathcal{L}(\gamma) </math> === Length for continuously differentiable paths === Let <math>\gamma : [a,b] \to \mathbb{C} </math> be a continuously differentiable path, then <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt </math></center> gives the length of the path <math>\gamma</math>. === Note - Length for continuously differentiable paths === Since <math>\gamma</math> is continuously differentiable, <math>\gamma{\,}'</math> is a continuous function. Since <math>[a,b]</math> is a compact interval, <math>\gamma{\,}'</math> takes a minimum and maximum. Therefore, <math>\gamma{\,}'</math> and <math>|\gamma{\,}'|</math> are bounded, and we have: <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt < \infty </math></center> === Piecewise continuously differentiable curves === In general, piecewise <math>C^1</math>-curves are always rectifiable, because we can apply the above considerations to the individual parts of the curve, which then additively give the length of the entire curve. In the further course of complex analysis, paths (e.g. over the triangle edge) are considered that only possess the property of continuous differentiability in a piecewise manner, for which we can then still calculate the length as the sum of the arc lengths. == Non-rectifiable curve == As an example of a non-rectifiable curve, consider <math>\gamma\colon[0,1]\to \mathbb C</math>, <center><math> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center> === Continuity - continuous differentiability === First, <math>\gamma</math> is continuous and, on each interval <math>[\epsilon,1]</math>, even continuously differentiable. On these intervals, the length is given by <center><math> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> === Calculation of improper integral === For <math>\epsilon > 0</math>, this converges to <center><math> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center> so <math>\gamma</math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/rektifizierbare Kurve|rektifizierbare Kurve]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/rektifizierbare Kurve]] </noinclude> 2ciutc72wh9988k9xq36wp48fn5xprh 2691778 2691777 2024-12-13T11:52:37Z Eshaa2024 2993595 /* Path length for differentiable paths */ 2691778 wikitext text/x-wiki == Definition == Let <math>\gamma\colon [a,b]\to \mathbb C</math> be a continuous curve. It is called rectifiable if its length <center><math>\mathcal{L}(\gamma) := \sup\left\{ \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| \ \bigg|\ n \in \mathbb N, a \le t_0 < \ldots < t_n \le b \right\} </math></center> is finite, and <math>\mathcal{L}(\gamma)</math> is called the length of <math>\gamma</math>. === Approximation of path length by polygonal chain === The following image shows how a polygonal chain <math>P</math> can be used to approximate the length of a curve <math>\gamma</math>. [[File:Laenge kurve rektifizierbarkeit.png|350px|rectifiable curve - approximation of length by polygonal chain - created with Geogebra on Linux]] === Estimation of length === The length of the polygonal chain <math>P_n</math> underestimates the actual length of a rectifiable curve <math>\gamma</math>, i.e. <math>\mathcal{L}(P_n) \leq \mathcal{L}(\gamma)</math>. In general, <math>\mathcal{L}(P_n) < \mathcal{L}(\gamma)</math>. By applying the triangle inequality, we get <math> < </math> if the path's trace is not a line. == Path length for differentiable paths == If <math>\gamma</math> is continuously differentiable, then <math>\gamma</math> is rectifiable. Let <math>a \le t_0 < \ldots < t_n \le b</math>, then there exists[[w:en:mean value theorem|mean value theorem]] <math>\tau_i \in (t_{i-1}, t_i)</math> such that <center><math> \sum_{i=1}^n |\gamma(t_{i})-\gamma(t_{i-1})| = \sum_{i=1}^n |{\gamma'(\tau_i)}| \cdot (t_{i} - t_{i-1}) </math></center> === Riemann sum as length of polygonal chain === The right-hand side of the above equation for the polygonal chain is a Riemann sum for the integral <math>\int_a^b |\gamma{\,}'(t)|\,dt</math>. If we take the maximum of the interval widths <math> \max_{i\in\{1, \ldots , n\} } (t_{i}-t_{t-1}) </math> for <math>n</math> to infinity, the length of the polygonal chains <math>\mathcal{L}(P_n)</math> converges to the length of the path <math> \mathcal{L}(\gamma) </math> === Length for continuously differentiable paths === Let <math>\gamma : [a,b] \to \mathbb{C} </math> be a continuously differentiable path, then <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt </math></center> gives the length of the path <math>\gamma</math>. === Note - Length for continuously differentiable paths === Since <math>\gamma</math> is continuously differentiable, <math>\gamma{\,}'</math> is a continuous function. Since <math>[a,b]</math> is a compact interval, <math>\gamma{\,}'</math> takes a minimum and maximum. Therefore, <math>\gamma{\,}'</math> and <math>|\gamma{\,}'|</math> are bounded, and we have: <center><math> \mathcal{L}(\gamma) = \int_a^b |\gamma'(t)|\, dt < \infty </math></center> === Piecewise continuously differentiable curves === In general, piecewise <math>C^1</math>-curves are always rectifiable, because we can apply the above considerations to the individual parts of the curve, which then additively give the length of the entire curve. In the further course of complex analysis, paths (e.g. over the triangle edge) are considered that only possess the property of continuous differentiability in a piecewise manner, for which we can then still calculate the length as the sum of the arc lengths. == Non-rectifiable curve == As an example of a non-rectifiable curve, consider <math>\gamma\colon[0,1]\to \mathbb C</math>, <center><math> t \mapsto \left\{\begin{array}{ll} 0 & t= 0 \\ t +it\cos t^{-1} & t > 0\end{array}\right. </math></center> === Continuity - continuous differentiability === First, <math>\gamma</math> is continuous and, on each interval <math>[\epsilon,1]</math>, even continuously differentiable. On these intervals, the length is given by <center><math> \mathcal{L}(\gamma|_{[\epsilon,1]}) = \int_\epsilon^1 \left| 1 - \frac i{t}\sin t^{-1}\right|\,dt . </math></center> === Calculation of improper integral === For <math>\epsilon > 0</math>, this converges to <center><math> \int_0^1 \left(1 + \frac 1{t^2}\sin^2 t^{-1}\right)^{1/2}\,dt = \infty </math></center> so <math>\gamma</math> is not rectifiable. ==See also== * [[Complex Analysis/Lemma_of_Goursat|Lemma of Goursat]] [[Category:Complex Analysis]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare_Kurve Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/rektifizierbare Kurve|rektifizierbare Kurve]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve * Date: 12/11/2024 <span type="translate" src="Course:Complex_Analysis//rectifiable Curve" srclang="de" date="12/11/2024" time="12:01" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/rektifizierbare Kurve]] </noinclude> 4x6wwcy6tt28pvn30zpcl54nlsswoj9 0 317160 2691672 2691398 2024-12-12T18:14:27Z Eshaa2024 2993595 2691672 wikitext text/x-wiki The Goursat Lemma is an important intermediate result in the proof of the [[Cauchy Integral Theorem]]. It restricts the integration paths to triangles, and its proof is based on a [[Complex_Analysis/Lemma_of_Goursat_(Details)|geometrical subdivision argument]]. ==Statement== Let <math>D \subseteq \mathbb C</math> be a closed triangle, <math>G \supseteq D</math> open, and <math>f \colon U \to \mathbb C</math> holomorphic. Then, <math>\int_{\partial D} f(z), dz = 0.</math> ==Proof== Let <math>\Delta_0 := D</math>. We will inductively construct a sequence <math>(\Delta_n)_{n \ge 0}</math> with the following properties: <math>\Delta_n \subseteq \Delta_{n-1}</math> <math>\mathcal{L}(\partial \Delta_n) = 2^{-n}\mathcal{L}(\partial D)</math> (where <math>L</math> denotes the [[Complex_Analysis/Curve|length of a curve]]) <math>\left|\int_{\partial D} f(z), dz\right| \le 4^n\left|\int_{\partial \Delta_n} f(z), dz\right|</math> So, for some <math>n \ge 0</math>, suppose <math>\Delta_n</math> is already constructed. We subdivide <math>\Delta_n</math> by connecting the midpoints of the sides, creating four smaller triangles <math>\Delta_{n+1}^i</math>, <math>1 \le i \le 4</math>. Since the connections of the midpoints cancel each other out during integration, we have: <center><math>\begin{array}{rl} \displaystyle\left|\int_{\partial \Delta_n} f(z)\, dz\right| &= \displaystyle\left|\sum_{i=1}^4 \int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\ &\le \displaystyle\sum_{i=1}^4 \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\ &\le \displaystyle\max_i \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right| \end{array}</math></center> Now, choose <math>1 \le i \le 4</math> with <math>\left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right| = \max_i\left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|</math> and set <math>\Delta_{n+1} := \Delta_{n+1}^i</math>. Then, by construction, we have <math>\Delta_{n+1}\subseteq \Delta_n</math>, and also: <center><math> \mathcal{L}(\partial \Delta_{n+1}) = \frac 12 \mathcal{L}(\partial \Delta_n) = 2^{-(n+1)} \mathcal{L}(\partial D) </math></center> and <center><math> \left|\int_{\partial D} f(z)\, dz\right| \le 4^n\left|\int_{\partial \Delta_{n}} f(z)\, dz\right| \le 4^{n+1} \left|\int_{\partial \Delta_{n+1}} f(z)\, dz\right| </math></center> Thus, <math>\Delta_{n+1}</math> has exactly the desired properties. Since all <math>\Delta_n</math> are compact, the intersection <math>\bigcap_{n\ge 0} \Delta_n \ne \emptyset</math>, and let <math>z_0 \in \bigcap_{n\ge 0} \Delta_n</math>. Since <math>f</math> is holomorphic at <math>z_0</math>, there exists a continuous function <math>A \colon V \to \mathbb C</math> with <math>A(z_0) = 0</math> in a neighborhood <math>V</math> of <math>z_0</math> such that: <center><math> f(z) = f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0), \qquad z \in V</math></center> Since <math>z \mapsto f(z_0) + (z-z_0)f'(z_0)</math> has an antiderivative, for all <math>n \ge 0</math> with <math>\Delta_n \subseteq V</math>, we have: <center><math> \int_{\partial \Delta_{n}} f(z)\, dz = \int_{\partial \Delta_n} f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0) \, dz = \int_{\partial \Delta_n} A(z)(z-z_0) \, dz. </math></center> Thus, using the continuity of <math>A</math> and <math>A(z_0) = 0</math>, we get: <center><math> \begin{array}{rl} \displaystyle\left|\int_{\partial D} f(z)\, dz\right| &\le \displaystyle 4^n\left|\int_{\partial \Delta_{n}} f(z)\, dz\right|\\ &= \displaystyle 4^n\left|\int_{\partial \Delta_{n}} A(z)(z-z_0)\, dz\right|\\ &\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n) \max_{z\in \partial \Delta_n} |z-z_0||A(z)|\\ &\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n)^2 \max_{z\in \partial \Delta_n} |A(z)|\\ &=\displaystyle \mathcal{L}(\partial D) \max_{z\in \partial \Delta_n} |A(z)|\\ &\to\displaystyle \mathcal{L}(\partial D) |A(z_0)| = 0, \qquad n \to \infty. \end{array}</math></center>==Notation in the Proof== <math>\Delta_n</math> is the <math>n</math>-th similar subtriangle of the original triangle with side lengths shortened by a factor of <math>\frac{1}{2^n}</math>. <math>\partial\Delta_n</math> is the integration path along the boundary of the <math>n</math>-th similar subtriangle, with a perimeter <math>\mathcal{L}(\partial\Delta_n) = \frac{1}{2^n}\cdot \mathcal{L}(\partial\Delta_0) </math>. ==See also== [[Complex_Analysis/Lemma_of_Goursat_(Details)|Goursat's Lemma with Details]] [[Complex_Analysis/Rectifiable_Curve|rectifiable curve or length of a curve]] [[Category:Complex_Analysis]] f2busdv8p7o6as84r4m2dwapt12fkgf 0 317168 2691765 2691481 2024-12-13T08:27:05Z Dan Polansky 33469 2691765 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at the concept of value. One inspiration is a note by Mandelbrot in his ''The (mis)behavior of Markets'' to the effect that the concept of value is mysterious, elusive or something of the sort. Another inspiration is the concept of Quality by Robert Pirsig, which Pirsig wants to leave undefined, but is somehow relating to goodness or value. I suspect that the resulting analysis is going to be weak and confusing. Perhaps Mandelbrot is right: the concept is elusive. Let me start with an initial take at a definition of value: * The degree to which someone is unwilling to part with something. The something one may want to part with can be an object but also a state of affairs or being. The concept of being used here is perhaps somewhat untraditional, so let me elucidate a bit. Someone who lives is London has ''being an entity living in London'' or ''being a London inhabitant''. If they move from London to, say, Edinburgh, they part with ''being a London inhabitant'' or with London-inhabitanthood. One may be unwilling to leave London, that is to say, to part with being a London inhabitant. And that degree of unwillingness is identical to the (subjective) value of that being. Havings are species of beings; that is, having is car is being such that one has a car. in this way, we may reduce all partings to parting with beings. Thus, instead of parting with a car, one parts with being one who has a car. This seems strangely technical, but it seems to have a theoretical or analytical significance or advantage. For one thing, one may part with an object only in part; for instance, a real estate owner renting the real estate does not part with being the rightful owner but does part with being the rightful user of the real estate. This all seems perhaps rather counter-intuitive and I hope to develop these ideas further in more detail later. The definition is doubly subjective. For one thing, the value of insulin for someone with diabetes will be different from its value to someone without. But additionally, someone with diabetes may happen to be in an unsound state of mind and be eager to part with insulin despite its obvious utility. The definition seems to fail to differentiate value from an ''estimate'' of value. Since, appreciation of something is limited by its knowledge. If I have a box with unknown content, the value of the box depends on the value of the unknown content. Learning about the content would change my unwillingness to part with the box (and the content), but it seems strange that the value would change only based on change in knowledge of the valuing person. The definition is made in terms of parting rather than acquisition. Whether it is the best option is unclear. At least, there are some things one does not acquire, e.g. hands. And for them, the relevant change is parting. Moreover, refusal to part generally does not require expenditure of attention, unlike a decision to acquire. == Use value vs. exchange value == The unwillingness to part may be driven by use value or by exchange value. Thus, someone with diabetes has use for insulin. By contrast, someone without diabetes does not has use for it, but can sell insulin or offer it to someone as a favor. Both cases match the definition, and this split seems to be meaningful. == Value vs. goodness == Goodness and value are two distinct concepts. A good knife may lose its value because of reduced demand on knifes, which may result from availability of cheap substitutes on the market. But the knife does not become any less good (in the sense of e.g. rust resistant, maintaining sharpness for long, etc.). Value has something to do with supply and demand. If the example of knife seems too contrived, one may think of factory machines that lose value as a result for loss of demand on the kind of product they are used to make or as a result of arrival of new technology requiring new machines. One can nonetheless perhaps connect value to goodness as follows. The value of the knife is given by the degree of goodness of having the knife. Here, goodness of the knife is distinguished from the goodness of ''having'' (a good) knife. With changing market conditions, the goodness of the knife does not change, but the goodness of having the knife does change. One may object that this departs from the doubly subjective definition by an implied objectivity of the word good. This would need to be clarified. For one thing, the idea that the goodness of having insulin at disposal differs between a person with diabetes and one without is plausible enough; so far so good. == Value vs. price == A thing or state of affairs does not need to be for sale to have value. For instance, having healthy organs is of value in a country that does not allow people to sell their body parts for money. Moreover, having the purchased food at home rather than in the grocery store is of additional value since this is where one wants to have the food. This is an example of not only having the food but also having it in the right place being of value. In some contexts, the value does seem to match the price. It is so at least linguistically: one talks about the value of the stock rising. And in so far as one mostly holds stock for the exchange value, the value is given by the price, although there is also the dividend income. == Objective value == I would like to come up with something like objective value or intersubjective value. Alas, there is some difficulty. One difficulty is something like inversion of value in transactions. Thus, when I buy e.g. bread for money, I indicate that the bread has more value to me than the money I give up. But for the seller, the relation has to be the opposite: the money must be of more value than the bread. Nonetheless, something like objective value in market context makes sense. Thus, I may have a certain degree of unwillingness to part with money in the given market conditions. That unwillingness does not seem to be purely subjective: whatever I want to acquire, I will need money for that, so I naturally do not freely want to part with the money without gaining something in exchange. And this applies to other people as well. Sure enough, the unwillingness to part with a specific amount of money is different between a rich person and a poor person. Be it as it may, one may reason as follows. There is more objective value in food, clothing and shelter since everyone needs those. There is less objective value is luxury items or various pieces of art whose trading resembles trading of bitcoin in certain ways. Let us consider: did the Tulips during the noted speculative bubble have objective value? They seemed to since they had exchange value. That is to say, a holder of a tulip could exchange it for something of value and in fact something of high value during the peak of the fever. This is strange. This kind of concept of objective value does not seem to be anything like underlying value, whatever that would be. But excluding exchange value from the concept of value does not seem reasonable. A puzzle. == Personal value == Things can have nostalgic or other psychologically subjective or personal value. Someone may grow fond of old items and be unwilling to part with them as a result, regardless of availability of cheap new replacement items on the market. Someone may appreciate old letters from a friend or a romantic partner, and these same letters will have almost no value for anyone else. == Marginalism == Marginalism feartures the interesting idea of additional items having increasingly less value. Thus, additional items of food have increasingly less use value; when one is satiated, one cannot appreciate the additional food items so much. == Objective value 2 == An attempt at objective value: * The degree to which something contributes to the continuing existence of the value assignor. This is very generic. The value assignor can be a human, an animal or a robot; perhaps even a plant. There is an element of objectivity: if someone who has diabetes does not appreciate insulin, he is objectively wrong about its value. How this concept is going to interact with exchange value is unclear. It should probably involve it as part: money has this kind of value--indirectly--for its ability to purchase things of direct value, e.g. food or drinking water. In a biological context, one may replace the objective of the continuing existence of the value assignor with the success of the genes of the value assignor. How this would translate into the case of robots would need to be figured out. In this conception of value, there is no room for subjective preferences. It seems to be a very cold, as if Spartan concept. We may modify the definition to be not in relation to continuing existence but to something like power or ability. Since, without modification, someone who is very rich does not gain much in terms of survival ability by earning additional huge sums of money, but it is implausible that these additional sums are of almost no value. == Things having value == At first thought, one may think that what is of value are objects, e.g. a loaf of bread or a silver coin. But it turns out that what is of value are beings (including havings), states of affairs. Even to speak of bread as having value seems to be a shorcut for saying that ''having bread'' is of value rather than e.g. its existence. And then, it is of more value to have the bread where one needs it rather than merely owning it; one spends effort to gain having the bread in the paltry. Other beings and havings of value include having hands, having both kidneys, having good friends and associates, having a good reputation, having an educational certificate, having skills even if uncertified, etc. Sometimes it is ''not'' having something that is of value, e.g. in a game mentioned by László Mérő in which one of the truck drivers throws away the steering wheel and thereby wins the game (detail to be added). Also of value may be negative objects, such as holes, lacks of fullness, etc. Thus, having pantry not completely full may be of value, or else one has no place for adding more items. Thus, getting rid of unnecessary items may be of value. For workers, being subjected to state-enacted work safety regulations may be of value; it improves their negotiation power. Here, the reduction of freedom of contract is of value. For some of the poorest people, being in a country such as the U.K. that has NHS (state-funded single-payer healthcare) may be of value. == Labor theory of value == This theory states that things have value to the degree of the amount of labor necessary to produce or acquire them. I find this theory wrong. It disregards the demand part: someone having spent a lot of labor in creating a useless thing does not automatically make the thing valuable. Furthermore, it disregards scarcity-driven value of natural resources. Sure enough, human labor is one key scarce resource affecting prices, resource most people have at disposal. But I see no reason why scarcity of other scarce resources should not affect prices or that it should not be reflected in the concept of value. Another objection is that labor by an unskilled person is generally not necessarily as valuable as labor by skilled person. One could try to account for this by considering the skill to be a result of past labor, effort spent in acquiring the skill. But acquiring the innate talent, a capital asset, does not seem to be a result of labor; it is not acquired at all, it seems. Further reading: * {{W|Labor theory of value}}, wikipedia.org == The value of art == One may think the value of art has something to do with immediate utility or quasi-utility, which, in the case of paintings, would consist in production of pleasing aesthetic response as a result of looking at the painting. Alas, that does not match art prices. There seems to be some strange speculative element in art prices. A key property of art items is that they are not so easy to copy or copying is prohibited. The items share this property with money. There is a scarcity of talented painters or there is a scarcity of positive critical appraisals for paintings by respected critics. What kind of game is really going on I do not know. Some people apparently take pleasure in destroying property to demonstrate wealth. Why, then, would one not want to demonstrate wealth by buying arguably overpriced fundamentally useless items? Look how wealthy I am, I can afford to spend this incredible amount of money for something of no apparent utility, at least as narrowly understood. Conspicuous consumption is the keyword. IEP takes a different view of the value of art. Further reading: * [https://iep.utm.edu/value-of-art/ Value of Art], Internet Encyclopedia of Philosophy == Further reading == * [[Wikisource: The measure of value stated and illustrated]] by Malthus * [[Wikisource: 1911 Encyclopædia Britannica/Value]] * [[Wikisource: The New International Encyclopædia/Value (political economy)]] * [https://plato.stanford.edu/entries/value-theory/ Value Theory], Stanford Encyclopedia of Philosophy -- despite the word "value", this seems to be about something else 7jxppil8hjq9za3b95gsrshztx8ka7t 2691767 2691765 2024-12-13T08:45:34Z Dan Polansky 33469 2691767 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at the concept of value. One inspiration is a note by Mandelbrot in his ''The (mis)behavior of Markets'' to the effect that the concept of value is mysterious, elusive or something of the sort. Another inspiration is the concept of Quality by Robert Pirsig, which Pirsig wants to leave undefined, but is somehow relating to goodness or value. I suspect that the resulting analysis is going to be weak and confusing. Perhaps Mandelbrot is right: the concept is elusive. Let me start with an initial take at a definition of value: * The degree to which someone is unwilling to part with something. The something one may want to part with can be an object but also a state of affairs or being. The concept of being used here is perhaps somewhat untraditional, so let me elucidate a bit. Someone who lives is London has ''being an entity living in London'' or ''being a London inhabitant''. If they move from London to, say, Edinburgh, they part with ''being a London inhabitant'' or with London-inhabitanthood. One may be unwilling to leave London, that is to say, to part with being a London inhabitant. And that degree of unwillingness is identical to the (subjective) value of that being. Havings are species of beings; that is, having is car is being such that one has a car. in this way, we may reduce all partings to parting with beings. Thus, instead of parting with a car, one parts with being one who has a car. This seems strangely technical, but it seems to have a theoretical or analytical significance or advantage. For one thing, one may part with an object only in part; for instance, a real estate owner renting the real estate does not part with being the rightful owner but does part with being the rightful user of the real estate. This all seems perhaps rather counter-intuitive and I hope to develop these ideas further in more detail later. Be it as it may, it is still meaningful to talk about the value of an object (a car, a golden piece, etc.) as a shortcut for the value of ''having'' the object (more technically, being the rightful owner of the object or other form of having). The definition is doubly subjective. For one thing, the value of insulin for someone with diabetes will be different from its value to someone without. But additionally, someone with diabetes may happen to be in an unsound state of mind and be eager to part with insulin despite its obvious utility. The definition seems to fail to differentiate value from an ''estimate'' of value. Since, appreciation of something is limited by its knowledge. If I have a box with unknown content, the value of the box depends on the value of the unknown content. Learning about the content would change my unwillingness to part with the box (and the content), but it seems strange that the value would change only based on change in knowledge of the valuing person. The definition is made in terms of parting rather than acquisition. Whether it is the best option is unclear. At least, there are some things one does not acquire, e.g. hands. And for them, the relevant change is parting. Moreover, refusal to part generally does not require expenditure of attention, unlike a decision to acquire. == Use value vs. exchange value == The unwillingness to part may be driven by use value or by exchange value. Thus, someone with diabetes has use for insulin. By contrast, someone without diabetes does not has use for it, but can sell insulin or offer it to someone as a favor. Both cases match the definition, and this split seems to be meaningful. == Value vs. goodness == Goodness and value are two distinct concepts. A good knife may lose its value because of reduced demand on knifes, which may result from availability of cheap substitutes on the market. But the knife does not become any less good (in the sense of e.g. rust resistant, maintaining sharpness for long, etc.). Value has something to do with supply and demand. If the example of knife seems too contrived, one may think of factory machines that lose value as a result for loss of demand on the kind of product they are used to make or as a result of arrival of new technology requiring new machines. One can nonetheless perhaps connect value to goodness as follows. The value of the knife is given by the degree of goodness of having the knife. Here, goodness of the knife is distinguished from the goodness of ''having'' (a good) knife. With changing market conditions, the goodness of the knife does not change, but the goodness of having the knife does change. One may object that this departs from the doubly subjective definition by an implied objectivity of the word good. This would need to be clarified. For one thing, the idea that the goodness of having insulin at disposal differs between a person with diabetes and one without is plausible enough; so far so good. == Value vs. price == A thing or state of affairs does not need to be for sale to have value. For instance, having healthy organs is of value in a country that does not allow people to sell their body parts for money. Moreover, having the purchased food at home rather than in the grocery store is of additional value since this is where one wants to have the food. This is an example of not only having the food but also having it in the right place being of value. In some contexts, the value does seem to match the price. It is so at least linguistically: one talks about the value of the stock rising. And in so far as one mostly holds stock for the exchange value, the value is given by the price, although there is also the dividend income. == Objective value == I would like to come up with something like objective value or intersubjective value. Alas, there is some difficulty. One difficulty is something like inversion of value in transactions. Thus, when I buy e.g. bread for money, I indicate that the bread has more value to me than the money I give up. But for the seller, the relation has to be the opposite: the money must be of more value than the bread. Nonetheless, something like objective value in market context makes sense. Thus, I may have a certain degree of unwillingness to part with money in the given market conditions. That unwillingness does not seem to be purely subjective: whatever I want to acquire, I will need money for that, so I naturally do not freely want to part with the money without gaining something in exchange. And this applies to other people as well. Sure enough, the unwillingness to part with a specific amount of money is different between a rich person and a poor person. Be it as it may, one may reason as follows. There is more objective value in food, clothing and shelter since everyone needs those. There is less objective value is luxury items or various pieces of art whose trading resembles trading of bitcoin in certain ways. Let us consider: did the Tulips during the noted speculative bubble have objective value? They seemed to since they had exchange value. That is to say, a holder of a tulip could exchange it for something of value and in fact something of high value during the peak of the fever. This is strange. This kind of concept of objective value does not seem to be anything like underlying value, whatever that would be. But excluding exchange value from the concept of value does not seem reasonable. A puzzle. == Personal value == Things can have nostalgic or other psychologically subjective or personal value. Someone may grow fond of old items and be unwilling to part with them as a result, regardless of availability of cheap new replacement items on the market. Someone may appreciate old letters from a friend or a romantic partner, and these same letters will have almost no value for anyone else. == Marginalism == Marginalism feartures the interesting idea of additional items having increasingly less value. Thus, additional items of food have increasingly less use value; when one is satiated, one cannot appreciate the additional food items so much. == Objective value 2 == An attempt at objective value: * The degree to which something contributes to the continuing existence of the value assignor. This is very generic. The value assignor can be a human, an animal or a robot; perhaps even a plant. There is an element of objectivity: if someone who has diabetes does not appreciate insulin, he is objectively wrong about its value. How this concept is going to interact with exchange value is unclear. It should probably involve it as part: money has this kind of value--indirectly--for its ability to purchase things of direct value, e.g. food or drinking water. In a biological context, one may replace the objective of the continuing existence of the value assignor with the success of the genes of the value assignor. How this would translate into the case of robots would need to be figured out. In this conception of value, there is no room for subjective preferences. It seems to be a very cold, as if Spartan concept. We may modify the definition to be not in relation to continuing existence but to something like power or ability. Since, without modification, someone who is very rich does not gain much in terms of survival ability by earning additional huge sums of money, but it is implausible that these additional sums are of almost no value. == Things having value == At first thought, one may think that what is of value are objects, e.g. a loaf of bread or a silver coin. But it turns out that what is of value are beings (including havings), states of affairs. Even to speak of bread as having value seems to be a shorcut for saying that ''having bread'' is of value rather than e.g. its existence. And then, it is of more value to have the bread where one needs it rather than merely owning it; one spends effort to gain having the bread in the paltry. Other beings and havings of value include having hands, having both kidneys, having good friends and associates, having a good reputation, having an educational certificate, having skills even if uncertified, etc. Sometimes it is ''not'' having something that is of value, e.g. in a game mentioned by László Mérő in which one of the truck drivers throws away the steering wheel and thereby wins the game (detail to be added). Also of value may be negative objects, such as holes, lacks of fullness, etc. Thus, having pantry not completely full may be of value, or else one has no place for adding more items. Thus, getting rid of unnecessary items may be of value. For workers, being subjected to state-enacted work safety regulations may be of value; it improves their negotiation power. Here, the reduction of freedom of contract is of value. For some of the poorest people, being in a country such as the U.K. that has NHS (state-funded single-payer healthcare) may be of value. == Labor theory of value == This theory states that things have value to the degree of the amount of labor necessary to produce or acquire them. I find this theory wrong. It disregards the demand part: someone having spent a lot of labor in creating a useless thing does not automatically make the thing valuable. Furthermore, it disregards scarcity-driven value of natural resources. Sure enough, human labor is one key scarce resource affecting prices, resource most people have at disposal. But I see no reason why scarcity of other scarce resources should not affect prices or that it should not be reflected in the concept of value. Another objection is that labor by an unskilled person is generally not necessarily as valuable as labor by skilled person. One could try to account for this by considering the skill to be a result of past labor, effort spent in acquiring the skill. But acquiring the innate talent, a capital asset, does not seem to be a result of labor; it is not acquired at all, it seems. Further reading: * {{W|Labor theory of value}}, wikipedia.org == The value of art == One may think the value of art has something to do with immediate utility or quasi-utility, which, in the case of paintings, would consist in production of pleasing aesthetic response as a result of looking at the painting. Alas, that does not match art prices. There seems to be some strange speculative element in art prices. A key property of art items is that they are not so easy to copy or copying is prohibited. The items share this property with money. There is a scarcity of talented painters or there is a scarcity of positive critical appraisals for paintings by respected critics. What kind of game is really going on I do not know. Some people apparently take pleasure in destroying property to demonstrate wealth. Why, then, would one not want to demonstrate wealth by buying arguably overpriced fundamentally useless items? Look how wealthy I am, I can afford to spend this incredible amount of money for something of no apparent utility, at least as narrowly understood. Conspicuous consumption is the keyword. IEP takes a different view of the value of art. Further reading: * [https://iep.utm.edu/value-of-art/ Value of Art], Internet Encyclopedia of Philosophy == Further reading == * [[Wikisource: The measure of value stated and illustrated]] by Malthus * [[Wikisource: 1911 Encyclopædia Britannica/Value]] * [[Wikisource: The New International Encyclopædia/Value (political economy)]] * [https://plato.stanford.edu/entries/value-theory/ Value Theory], Stanford Encyclopedia of Philosophy -- despite the word "value", this seems to be about something else h1guyu4vvnj1k52fo28o1nwat0oi7y9 0 317185 2691633 2691594 2024-12-12T14:24:29Z Dan Polansky 33469 2691633 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soliders are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude _CALC_. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] bimw3oe6vzpc6amfsfgusvz8crrgnqs 2691635 2691633 2024-12-12T14:43:17Z Dan Polansky 33469 2691635 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soliders are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude _CALC_. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. Wikipedia article on modal logics has a lattice structure of different axiomatic modal logics. One can ask which of these logics is the true valid one and why. This remains something of a puzzle. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. One may wonder what value can there be in mathematical first-order logic. Since, in order to execute the study, one needs to assume something like informal logic anyway. Thus, to prove theorems that are part of mathematical logic, one needs something like informal logic. We already know how informal logic works before we even started, so why the fuzz? What's the point of this bureaucratic exercise, investigating something that was clear before we even started? Not so fast. For one thing, it is on the basis of the mathematical formal symbolic logic that we can get such results as Gödel's incompleteness theorems. Without formalizing logic in this way to be studied as an object, it is unclear how these could be oobtained. Moreover, first-order logic opens itself directly to computer support, such as in theorem provers. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] 9dq3n3peznpz1b93ta08q82udy4mbmg 2691636 2691635 2024-12-12T14:47:21Z Dan Polansky 33469 2691636 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soliders are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude _CALC_. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. Wikipedia article on modal logics has a lattice structure of different axiomatic modal logics. One can ask which of these logics is the true valid one and why. This remains something of a puzzle. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. One may wonder what value can there be in mathematical first-order logic. Since, in order to execute the study, one needs to assume something like informal logic anyway. Thus, to prove theorems that are part of mathematical logic, one needs something like informal logic. We already know how informal logic works before we even started, so why the fuzz? What's the point of this bureaucratic exercise, investigating something that was clear before we even started? Not so fast. For one thing, it is on the basis of the mathematical formal symbolic logic that we can get such results as Gödel's incompleteness theorems. Without formalizing logic in this way to be studied as an object, it is unclear how these could be oobtained. Moreover, first-order logic opens itself directly to computer support, such as in theorem provers. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://plato.stanford.edu/search/searcher.py?query=logic Search for "logic" in Stanford Encyclopedia of Philosophy -- shows there are many articles on the subject * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] 96fx5opus7xwnj53l8o6kol878mnoeh 2691637 2691636 2024-12-12T14:47:59Z Dan Polansky 33469 /* Further reading */ 2691637 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soliders are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude _CALC_. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. Wikipedia article on modal logics has a lattice structure of different axiomatic modal logics. One can ask which of these logics is the true valid one and why. This remains something of a puzzle. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. One may wonder what value can there be in mathematical first-order logic. Since, in order to execute the study, one needs to assume something like informal logic anyway. Thus, to prove theorems that are part of mathematical logic, one needs something like informal logic. We already know how informal logic works before we even started, so why the fuzz? What's the point of this bureaucratic exercise, investigating something that was clear before we even started? Not so fast. For one thing, it is on the basis of the mathematical formal symbolic logic that we can get such results as Gödel's incompleteness theorems. Without formalizing logic in this way to be studied as an object, it is unclear how these could be oobtained. Moreover, first-order logic opens itself directly to computer support, such as in theorem provers. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://plato.stanford.edu/search/searcher.py?query=logic Search for "logic"] in Stanford Encyclopedia of Philosophy -- shows there are many articles on the subject * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] lb4d81hgg2u8d5cq8vw07w9iur0rosx 2691638 2691637 2024-12-12T14:53:42Z Dan Polansky 33469 2691638 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soldiers are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude that there are 48 soliders. One may then perhaps ask whether the whole of mathematics is a branch of logic. Whatever the case, there is a traditional separation of logic from mathematics. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. Wikipedia article on modal logics has a lattice structure of different axiomatic modal logics. One can ask which of these logics is the true valid one and why. This remains something of a puzzle. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. One may wonder what value can there be in mathematical first-order logic. Since, in order to execute the study, one needs to assume something like informal logic anyway. Thus, to prove theorems that are part of mathematical logic, one needs something like informal logic. We already know how informal logic works before we even started, so why the fuzz? What's the point of this bureaucratic exercise, investigating something that was clear before we even started? Not so fast. For one thing, it is on the basis of the mathematical formal symbolic logic that we can get such results as Gödel's incompleteness theorems. Without formalizing logic in this way to be studied as an object, it is unclear how these could be oobtained. Moreover, first-order logic opens itself directly to computer support, such as in theorem provers. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://plato.stanford.edu/search/searcher.py?query=logic Search for "logic"] in Stanford Encyclopedia of Philosophy -- shows there are many articles on the subject * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] 383jfpo7ld1lo2370i4sihbm93xm92m 2691639 2691638 2024-12-12T15:15:40Z Dan Polansky 33469 2691639 wikitext text/x-wiki {{Original research}} This article by Dan Polansky looks at logic, the study of correct inference. It is in part idiosyncratic. Let me open the discussion by asking why anyone would want to study correct inference, correct conclusion drawing, that is, production of correct/true statements from correct/true statements. Are we not all born with ability to draw conclusions from premises? Can express articulation of principles of correct inference really bring us forward in any way? My tentative answer is yes, studying correct inference is of value. Above all, our experience shows that humans are too frail, too ready to make errors in inference/conclusion drawing. Given this fact, it does not yet follow that logic is going to help. Whether logic is going to help is an empirical question in the field of human psychology; it cannot be answered purely logically. It could turn out that people who learn logic (especially formal logic) do not really improve in ability to draw correct conclusions. One kind of logic taught is propositional logic. Here one learns to interpret logical connectives (and, or, implication, not) as truth-value/boolean functions. Thus, one can think of them as algebraic operators defines by means of truth-value tables. The idea is of logic as algebra. One can ask whether this brings us any further. It does. For instance, in natural language, or is sometimes implied to mean exclusive or. By defining the logical or by means of a table, one removes all ambiguity. One says: in logic, when we say or, this is what we mean. Another important idea is to interpret sentences as propositions that have truth value, true or false. That assumes the law of the excluded middle: a sentence has to be either true or false (whatever our knowledge of it). It is not obvious that sentences in natural language generally can be unambiguously interpreted in that way. Proposition logic requires us to try to think of unambiguous sentences that have truth value; if a sentence is ambiguous, it cannot be immediately fed as an input into propositional logic. Another think of note is the table-based implication. It is defined as follows A ==> B =def= A or not B. One sometimes reads "==>" as implies or from which follows, but that does not really make sense. The idea that from an untrue state any true statement follows seems suspect. Thus, the idea that e.g. from the grass being always yellow it follows that all cars are green does not make sense. In case of doubt, one is well reminded that "==>" is defined by the truth table, which is equivalent to A or not B. The real powerhorse is the first-order predicate logic. It seems to be based on the 19th century work of Frege. Here, one adds variables and existential and universal quantifiers as well as predicate symbols and function symbols. The variable refers to entities in the universe of discourse, that is, entity one can talk about given the particular language of concern. A language of concern is a set of predicate and function symbols together with their arities; semantics is not involved. The "first-order" part in the name refers to the quantification being only over items in the universe of discourse and not over sets of such items. Natural language is sometimes said to be not logical. That is misleading. In fact, language cannot violate the laws of logic. What is often meant by it is that language contains a lot of peculiarities, deviations from pattern-based expectations. For instance, one could think that "here" and "where" would be pronounced is a similar way, but that is not so. More importantly, there are semantic peculiarities, in which the semantics deviates from the pattern-based expectation. None of this violates the canons of logic. One simply has to learn that instead of making pattern-based guesses/estimates, one has to get more serious about word and phrase meaning, examining the meaning of each indivudal item in case of doubt regardless of the suggestiveness of the morphology or etymology. We may also mention Aristotle. He pointed out that we can sometimes reliably produce true sentences from true sentences. Thus, we can in fact discover some purely mechanical rules. A classic example is this: Socrates is a human; all humans are mortal ==> Socrates is mortal. This reminds us of the predicate logic, but the Aristotelian logic is much less powerful. I will not delve more into this here since I find it mainly of historical interest; if one is serious about logic, one should go for the first-order predicate logic. Strangely enough, arithmetic calculation can be seen as a species of logic in that it is in the business of mechanically producing true sentences from true sentences. For instance, from noting that soldiers are in a rectangular formation of 6 rows and 8 columns, we may reliably conclude that there are 48 soliders. One may then perhaps ask whether the whole of mathematics is a branch of logic. Whatever the case, there is a traditional separation of logic from mathematics. One concern about application of logic is that in order to produce true sentences from true sentences, we need to have some true sentences to start with, obtained without use of logic. That is true enough; these can be observational report sentences. One can charge that the observational report sentences are uncertain, and therefore, also the strictly logical conclusions are uncertain. That may be true in principle, but does not really seem practically relevant. For instance, we think to know reliably that Socrates is a human and that all humans are mortal; and then, we feel comfortable about drawing the conclusion that Socrates is mortal. That said, the GIGO problem (garbage in, garbage out) is in general a real one as for mechanical/algorithmic sentence production. There are too many sentences that we do not know reliably enough and yet we want to draw correct conclusions. Importantly, the mechanical conclusion drawing is of great value as part of falsificationism: if an uncertain sentence has a necessary logical consequence known to be untrue, the sentence cannot be true. Rejection of mechanical deductive inference as a principle would seem to prevent falsificationism from operating. One idea brought forward by the first-order predicate logic is that mechanical rules work well when all symbols are unambigous. The mechanisms of this logic do not have any way to disambguate by context; all occurrences by a symbol (predicate, function or variable) are taken to mean the same thing. One suggestion is then that human mind is helped when sentences deliberated about have reduced ambiguity; something like the logical engines in the background mind can start to work much better. However, this is an empirical hypothesis and would need a proper examination. A related idea is something that I call export of semantic items on the syntactic surface. Formal symbolic logic can only operate on what has been expressly stated using syntactic means as part of a sentence. Human deliberation about sentences often does not work like that; practical conclusion drawing often involves incorporation of unstated assumptions. Symbolic logic can inspire us to state additional assumption to make purely mechanical inference and argument verification work. One interesting application of predicate logic is having the pronoun ''nothing'' disappear by translating sentences into their logical form. Thus, the sentence "there is nothing in the box" can be rendered as "for each macroscopic object, it holds that it is not in the box". This points to natural language syntactically constructing apparent objects that are in fact not there, as part of something like syntactic sugar, here the putative referent of the word ''nothing'' that is allegedly contained in the box. The syntactic sugar is nice to have; it is much nicer to say "there is nothing in the box" or respond to the question "what is in the box" with "nothing" than use the more complex phrasing used above. And then, one can suspect that inquiries into the so-called ''nothingness'' end up to be nonsense (or maybe not?). There are various specialized formal symbolic logics, e.g. modal and temporal logics. In modal logics, the formal operators are interrelated in the same way as existential and universal quantifiers: <> =def= not [] not. Wikipedia article on modal logics has a lattice structure of different axiomatic modal logics. One can ask which of these logics is the true valid one and why. This remains something of a puzzle. More is currently for further reading. Apart from formal symbolic logic, there is also a thing called informal logic. It investiages e.g. logical fallacies, a classic being ad hominem. It does seem to have the capacity to reduce the rate of certain kind of wrong arguments, but to what extent it really does is again an empirical question. There is also something called argumentation theory. One would think it could be part of logic. I would need to have a closer look at it to see what it does. As a first note, it appears clear that the ''support'' relation (a statement supporting another statement) is usually not necessary one of strict deductive inference. Something else must be going on, but what it is exactly I would need to figure out. One part of argumentation is something I would call argument and counter-argument, on nested level. It is reminiscent of Popper's conjectures and refutations and Lakatos' proofs and refutations, but it can be something somewhat different. One idea is that in order to critically investigate a statement, one must allow even relatively weak counter-arguments into the discussion (but not completely irrelevant). And then one may criticize the counter-arguments as well, leading to a nested argument structure. Wikidebates in Wikiversity are a great example of this structure. There is something called inductive logic. From what I remember, Popper says something to the effect that there is no such thing as inductive logic since logic is the study of correct inference and inductive inferences are not correct. I would like to look more into the matter, paying more attention to defenders of induction (Carnap?) I also need to clarify whether I want to treat of induction here or in the epistemology article. The relation of logic to epistemology should be clarified. Logic could be seen as part of epistemology; since, if someone asks me how do I know that Socrates is mortal, I can answer: I know it by applying mechanical rules of logical inference, taking reliably known facts as an input. Mathematical symbolic formal logic can be contrasted to logic used in mathematics by mathematicians. There is a certain degree of informality in mathematical proofs, even when they invoke existential and universal quantifiers. Mathematical logic sets up axioms and proofs (which it sometimes calls derivations) as formally mathematically concieved/defined entities, subject to rigorous mathematical analysis. And thus, mathematical logic is metamatematics (matematics about tools used by mathematics) as well as metafield (field about tools used by various fields of inquiry). Let us recall that mathematics was not in a very bad state before the arrival of Fregean logic in the 19th century. Mathematicians succeeded in doing mathematics at least since the Ancient Greek Euclid, noted for the axiomatic system of Euclidean geometry. It would seem that mathematicians must have informally known something like first-order predicate logic all along. Which really is the case I do not know; this would require a thorough and serious look into the history of mathematics. One could argue that Newton and Leibniz did not practice the modern mathematical rigor with their early versions of calculus and that therefore something could have changed with the arrival of mathematical logic, especially with Cantor's set theory. One would do well to investiage the possible impact of Frege on Cantor. One may wonder what value can there be in mathematical first-order logic. Since, in order to execute the study, one needs to assume something like informal logic anyway. Thus, to prove theorems that are part of mathematical logic, one needs something like informal logic. We already know how informal logic works before we even started, so why the fuzz? What's the point of this bureaucratic exercise, investigating something that was clear before we even started? Not so fast. For one thing, it is on the basis of the mathematical formal symbolic logic that we can get such results as Gödel's incompleteness theorems. Without formalizing logic in this way to be studied as an object, it is unclear how these could be oobtained. Moreover, first-order logic opens itself directly to computer support, such as in theorem provers. There are multi-valued mathematical logics, including fuzzy logic. Thus, instead of a predicate either being true or false about its subject or subjects, the truth value can have degrees. In fuzzy logic, the truth value (also interpreted as degree of memebership in a fuzzy set) is a real number in [0, 1]. One then has to figure out how to calculate logical connectives and, or, implication and not, and multiple proposals are given. Fuzzy logic has applications in devices such as photographic cameras. There is what is known as intuitionist logic. (One should not read too much into the name "intuitionist", I think. The intent does not seem to be to abandon formal rigor in favor of something like Poincaré's mathematical intution.) It is weaker than the classical logic (such as the classical first-order logic): it makes fewer inferences possible. There seems to be the idea of ''constructive'' involved. I know almost nothing about it; further reading can be in SEP and WP. One can sometimes read that logic is a normative field. I find that doubtful. Logic does not tell anyone how he ''ought'' to think or whether he has anything like a duty to think in a particular way. Logic says: if you want to avoid producing untrue statements from true statements, here is how to go about it. A society can in fact require people to adhere to canons of logic, but that is not because logic says it should. Similarly, bridge engineering studies parameters of bridges and manner of bridge building that lead to low likelihood of the bridge failing. Bridge engineering does not tell anyone that they have a duty to build good bridges. Thus, bridge engineering is not a normative field. And nor is logic. I do not find the idea of logic being normative entirely wrong, in part since indeed, if e.g. a judge openly violates canons of logic or sound reasoning, there is likely to be a complaint that he broke his duty. It is just that the putative duty to engage in correct reasoning can be separated from study of correct reasoning. Logic is sometimes contrasted to psychology of reasoning. Popper argues these are different fields or domains and I find that convincing. On one hand, people often do feel the force of logic as if it was innate (and perhaps it is in some sense). On the other hand, people in fact often do reason in incorrect or brutally heuristic ways; logic does not recognize that reasoning as valid only because it is or seems natural. Thus, logic does not seem to be part of psychology. It is this contrast that may lead people to say that logic is normative. But perhaps it is more debatable than seems to me. == Further reading == * {{W|Logic}}, wikipedia.org * [https://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy -- features first-order predicate logic * [https://plato.stanford.edu/search/searcher.py?query=logic Search for "logic"] in Stanford Encyclopedia of Philosophy -- shows there are many articles on the subject * [https://www.fi.muni.cz/usr/kucera/teaching/logic/log.pdf Matematická logika] by Antonín Kučera (in Czech) [[Category:Logic]] 7fsxmpcnb0vmha5ikfik0vjtk8sy53u 6 317189 2691616 2024-12-12T12:59:50Z Young1lim 21186 {{Information |Description=VLSI.Arith: Carry Skip Adders 1A (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691616 wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith: Carry Skip Adders 1A (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} nq97xk5clfn7yqp5l1azu8bj2j3f158 6 317190 2691625 2024-12-12T13:43:47Z Young1lim 21186 {{Information |Description=Laurent.5: Permutations 6B (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691625 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutations 6B (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} bxtazthci7da0me7beceu9f9latv194 6 317191 2691627 2024-12-12T13:46:39Z Young1lim 21186 {{Information |Description=C04.SA0: Address-of and de-reference operators 1A (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691627 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address-of and de-reference operators 1A (20241212 - 20241211) |Source={{own|Young1lim}} |Date=2024-12-12 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3wgfq71jp8gy14hdw8k6qwxoq9zv15b 0 317193 2691641 2024-12-12T15:28:13Z Lbeaumont 278565 Created the topic 2691641 wikitext text/x-wiki {{TOC right | limit|limit=2}} [[w:Grammatical_modifier|Modifiers]] are words, phrases, or clauses that provide additional information about another element in a sentence.<ref>[[w:ChatGPT|ChatGPT]] generated this text responding to the prompt: “Generate a tutorial on the topic of dangling modifiers. Describe what dangling modifiers are. Give several examples. Identify the ambiguity of each and provide a rewritten version that clarifies each example.”</ref> A [[w:Dangling_modifier|dangling modifier]] occurs when the modifier does not clearly or logically relate to the word it is intended to describe. This can create confusion or unintended meanings for the reader. == '''What Are Dangling Modifiers?''' == A dangling modifier is a descriptive phrase or clause that appears to modify the wrong noun or no noun at all, leaving the sentence ambiguous or nonsensical. == '''Key Characteristics of Dangling Modifiers:''' == # They are often found at the beginning of a sentence. # They leave the reader wondering what the modifier is supposed to describe. # They create unintended or humorous meanings when interpreted literally. == '''Examples and Clarifications''' == '''Example 1: The Confused Runner''' '''Dangling Modifier:''' Running through the park, the flowers were in full bloom. '''Ambiguity:''' Who or what is running through the park? This sentence suggests that the flowers are running. '''Rewritten Version:''' Running through the park, I noticed the flowers were in full bloom. '''Example 2: The Sleepy Student''' '''Dangling Modifier:''' After studying all night, the exam was surprisingly easy. '''Ambiguity:''' Who studied all night? The sentence implies that the exam did the studying. '''Rewritten Version:''' After studying all night, she found the exam surprisingly easy. '''Example 3: The Flying Homework''' '''Dangling Modifier:''' Flying through the air, the homework landed on the teacher's desk. '''Ambiguity:''' Who or what is flying through the air? It sounds as though the homework is flying by itself. '''Rewritten Version:''' As I threw it across the room, the homework landed on the teacher's desk. '''Example 4: The Forgotten Keys''' '''Dangling Modifier:''' Left on the table, I couldn’t find my keys. '''Ambiguity:''' Who or what was left on the table? The sentence suggests that the speaker was left on the table. '''Rewritten Version:''' I couldn’t find my keys because they were left on the table. '''Example 5: The Barking Dog''' '''Dangling Modifier:''' Barking loudly, the mailman avoided the dog. '''Ambiguity:''' Who or what is barking? It sounds as though the mailman is barking. '''Rewritten Version:''' The dog was barking loudly as the mailman avoided it. == '''Tips for Identifying and Fixing Dangling Modifiers''' == # '''Locate the Modifier:''' Identify the descriptive phrase or clause at the beginning or end of the sentence. # '''Identify the Subject:''' Ensure that the subject being described is explicitly stated immediately after the modifier. # '''Revise the Sentence:''' Rewrite the sentence so the modifier clearly relates to the intended subject. == '''Practice Exercises''' == Try revising the following sentences to eliminate dangling modifiers: # Driving down the road, the sunset was beautiful. # Walking into the room, a strange smell was noticed. # Forgotten by her friends, the party was a lonely affair. == '''Conclusion''' == Dangling modifiers can lead to confusion and unintended humor in writing. By ensuring that every modifier has a clear and logical subject, you can create sentences that are precise, clear, and effective. Practice identifying and revising dangling modifiers to improve your writing skills. 0zaj6b83kkhpxfcr5ountsrbj9m8bku 2691684 2691641 2024-12-12T20:12:53Z Lbeaumont 278565 Improved links 2691684 wikitext text/x-wiki {{TOC right | limit|limit=2}} [[w:Grammatical_modifier|Modifiers]] are words, phrases, or [[w:Clause|clauses]] that provide additional information about another element in a sentence.<ref>[[w:ChatGPT|ChatGPT]] generated this text responding to the prompt: “Generate a tutorial on the topic of dangling modifiers. Describe what dangling modifiers are. Give several examples. Identify the ambiguity of each and provide a rewritten version that clarifies each example.”</ref> A [[w:Dangling_modifier|dangling modifier]] occurs when the modifier does not clearly or logically relate to the word it is intended to describe. This can create confusion or unintended meanings for the reader. == '''What Are Dangling Modifiers?''' == A [[w:Dangling_modifier|dangling modifier]] is a descriptive phrase or clause that appears to modify the wrong noun or no noun at all, leaving the sentence ambiguous or nonsensical. == '''Key Characteristics of Dangling Modifiers:''' == # They are often found at the beginning of a sentence. # They leave the reader wondering what the modifier is supposed to describe. # They create unintended or humorous meanings when interpreted literally. == '''Examples and Clarifications''' == === '''Example 1: The Confused Runner''' === '''Dangling Modifier:''' Running through the park, the flowers were in full bloom. '''Ambiguity:''' Who or what is running through the park? This sentence suggests that the flowers are running. '''Rewritten Version:''' Running through the park, I noticed the flowers were in full bloom. === '''Example 2: The Sleepy Student''' === '''Dangling Modifier:''' After studying all night, the exam was surprisingly easy. '''Ambiguity:''' Who studied all night? The sentence implies that the exam did the studying. '''Rewritten Version:''' After studying all night, she found the exam surprisingly easy. === '''Example 3: The Flying Homework''' === '''Dangling Modifier:''' Flying through the air, the homework landed on the teacher's desk. '''Ambiguity:''' Who or what is flying through the air? It sounds as though the homework is flying by itself. '''Rewritten Version:''' As I threw it across the room, the homework landed on the teacher's desk. === '''Example 4: The Forgotten Keys''' === '''Dangling Modifier:''' Left on the table, I couldn’t find my keys. '''Ambiguity:''' Who or what was left on the table? The sentence suggests that the speaker was left on the table. '''Rewritten Version:''' I couldn’t find my keys because they were left on the table. === '''Example 5: The Barking Dog''' === '''Dangling Modifier:''' Barking loudly, the mailman avoided the dog. '''Ambiguity:''' Who or what is barking? It sounds as though the mailman is barking. '''Rewritten Version:''' The dog was barking loudly as the mailman avoided it. == '''Tips for Identifying and Fixing Dangling Modifiers''' == # '''Locate the Modifier:''' Identify the descriptive phrase or clause at the beginning or end of the sentence. # '''Identify the Subject:''' Ensure that the [[w:Subject_(grammar)|subject]] being described is explicitly stated immediately after the modifier. # '''Revise the Sentence:''' Rewrite the sentence so the modifier clearly relates to the intended subject. == '''Practice Exercises''' == Try revising the following sentences to eliminate dangling modifiers: # Driving down the road, the sunset was beautiful. # Walking into the room, a strange smell was noticed. # Forgotten by her friends, the party was a lonely affair. == '''Conclusion''' == Dangling modifiers can lead to confusion and unintended humor in writing. By ensuring that every modifier has a clear and logical subject, you can create sentences that are precise, clear, and effective. Practice identifying and revising dangling modifiers to improve your writing skills. 1b1j9x9xnmggncl7315f64azxgmh3ea 2 317194 2691643 2024-12-12T15:30:12Z MMCLXXII 2994915 this is my user intro page i think and i added stuff about me and my wiki usage 2691643 wikitext text/x-wiki hello world i am not ready to doxx myself yet and i am very new on wikipedia as an account user. i don't want my name to be red like that so i'm writing a bit here. i am mostly a watcher here and rarely get involved as i am not that informed about editing in wikimedia. if i made an edit, it was probably to fix the grammar issues. if i edit the content, that means it either annoyed me to death or you screwed up so bad i had to intervene. you may see me on the talk/discuss pages. that's because i love doing both :thumbsup: cheers, ds3fqm0dcgkrluhj1ghr8dt6os3wda6 1 317195 2691644 2024-12-12T15:36:08Z Lbeaumont 278565 /* Community Economics presentation */ new section 2691644 wikitext text/x-wiki == Community Economics presentation == Consider adapting materials from this [https://www.e-c-o.net/wiki/Econet/CommunityEconomicsPresentation Community Economics presentation] to improve this course. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:36, 12 December 2024 (UTC) nmg2hucm75te6c7kqqtzrn7jkw9rw96 2 317196 2691656 2024-12-12T16:46:12Z DerGeist4040 2994919 New resource with "DerGeist4040 is a middle school student, ⁣mainly interested in roman governments and military tactics. He is also interested in Greek mythology." 2691656 wikitext text/x-wiki DerGeist4040 is a middle school student, ⁣mainly interested in roman governments and military tactics. He is also interested in Greek mythology. o8jf2drb5ctt404fhn164rajryzq7z5 0 317197 2691669 2024-12-12T17:54:09Z Eshaa2024 2993595 New resource with "==Definition== A curve in <math>\mathbb C</math> is a continuous mapping <math>\gamma \colon [a,b] \to \mathbb C</math>. The image set <center><math> \mathrm{Trace}(\gamma) := \{\gamma(t): t \in [a,b]\} </math></center> is called the "trace" of the curve. ===Properties of Curves=== * A curve is called differentiable, <math>C^k</math>, etc., if the defining mapping has these properties. *A property particularly important for the theory oCourse:Complex Analysis/Curve in..." 2691669 wikitext text/x-wiki ==Definition== A curve in <math>\mathbb C</math> is a continuous mapping <math>\gamma \colon [a,b] \to \mathbb C</math>. The image set <center><math> \mathrm{Trace}(\gamma) := \{\gamma(t): t \in [a,b]\} </math></center> is called the "trace" of the curve. ===Properties of Curves=== * A curve is called differentiable, <math>C^k</math>, etc., if the defining mapping has these properties. *A property particularly important for the theory o[[Course:Complex Analysis/Curve integrals|Curve integrals]] is the [[Course:Complex Analysis/rectifiable curve|rectifiable curve]]rectifiable of a curve, rectifiable curves are curves over which integration is possible. ==Page Information== This learning resource can be displayed as a Wiki2Reveal slide deck. === Wiki2Reveal === This Wiki2Reveal slide deck was created for the learning unit Course: Function Theory. The link for the Wiki2Reveal slides was generated using the Wiki2Reveal link generator. <!-- * The content of the page is based on the following sources: ** [https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve] -->This page was created as a document type PanDocElectron-SLIDE. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve. See also more information on Wiki2Reveal and the Wiki2Reveal link generator. <!-- * Next content in the course is [[]] -->[[Category:Wiki2Reveal]] jxteokwtvffcxwgdecr7e1olsnyjz86 2691776 2691669 2024-12-13T11:07:53Z Eshaa2024 2993595 2691776 wikitext text/x-wiki ==Definition== A curve in <math>\mathbb C</math> is a continuous mapping <math>\gamma \colon [a,b] \to \mathbb C</math>. The image set <center><math> \mathrm{Trace}(\gamma) := \{\gamma(t): t \in [a,b]\} </math></center> is called the "trace" of the curve. ===Properties of Curves=== * A curve is called differentiable, <math>C^k</math>, etc., if the defining mapping has these properties. *A property particularly important for the theory of [[Complex Analysis/curve integrals|curve integrals]] is the[[Complex Analysis/rectifiable curve|rectifiable curve]] rectifiable of a curve, rectifiable curves are curves over which integration is possible. ==Page Information== This learning resource can be displayed as a Wiki2Reveal slide deck. === Wiki2Reveal === This Wiki2Reveal slide deck was created for the learning unit Course: Function Theory. The link for the Wiki2Reveal slides was generated using the Wiki2Reveal link generator. <!-- * The content of the page is based on the following sources: ** [https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve] -->This page was created as a document type PanDocElectron-SLIDE. Link to the source on Wikiversity: https://de.wikiversity.org/wiki/%20Kurs:Funktionentheorie/Kurve. See also more information on Wiki2Reveal and the Wiki2Reveal link generator. <!-- * Next content in the course is [[]] -->[[Category:Wiki2Reveal]] l2seoec8qy9smsfa5p8rweb10a2ks9g 3 317198 2691674 2024-12-12T18:18:16Z Dan Polansky 33469 New resource with "== Course:Complex Analysis/Curve == You created page [[Course:Complex Analysis/Curve]], starting with "Course:", which I find unexpected. There is [[Complex Analysis/Curve]] page already. What are you trying to do with the page? --~~~~" 2691674 wikitext text/x-wiki == Course:Complex Analysis/Curve == You created page [[Course:Complex Analysis/Curve]], starting with "Course:", which I find unexpected. There is [[Complex Analysis/Curve]] page already. What are you trying to do with the page? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:18, 12 December 2024 (UTC) s1dtfv0z5d4ar2wsd13h34jtdnm69lm 2691675 2691674 2024-12-12T18:18:35Z Dan Polansky 33469 2691675 wikitext text/x-wiki == Course:Complex Analysis/Curve == You created page [[Course:Complex Analysis/Curve]], starting with "Course:", which I find unexpected. There is [[Complex Analysis/Curves]] page already. What are you trying to do with the page? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:18, 12 December 2024 (UTC) jzyjy2jh05erhmam04jakxrs0rblvhv 3 317199 2691676 2024-12-12T18:19:41Z Atcovi 276019 /* Welcome */ new section 2691676 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], DerGeist4040!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. 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See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:19, 12 December 2024 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} t5d157qk0pfqtcgqxr2j6a1bg8qqg2l 1 317201 2691679 2024-12-12T19:22:35Z DerGeist4040 2994919 /* Definition of "Pedophilia" */ new section 2691679 wikitext text/x-wiki == Definition of "Pedophilia" == I am a bit confused about the definition this article gives. It says that to be a pedophile, you must have sexual interest in any prepubescent child. Would that mean anyone who likes a child older than 13 is not a pedophile? [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 19:22, 12 December 2024 (UTC) kxfvl7y2s27b41vh5fg2dt3ukh3w495 2691707 2691679 2024-12-12T22:58:12Z Atcovi 276019 /* Definition of "Pedophilia" */ Reply 2691707 wikitext text/x-wiki == Definition of "Pedophilia" == I am a bit confused about the definition this article gives. It says that to be a pedophile, you must have sexual interest in any prepubescent child. Would that mean anyone who likes a child older than 13 is not a pedophile? [[User:DerGeist4040|DerGeist4040]] ([[User talk:DerGeist4040|discuss]] • [[Special:Contributions/DerGeist4040|contribs]]) 19:22, 12 December 2024 (UTC) :{{ping|DerGeist4040}} Correct as pedophiles are attracted to prepubescent children. And take note that the definition of pedophilia here is being given from a notable source (https://www.annualreviews.org/doi/10.1146/annurev.clinpsy.032408.153618), and is not the author's definition. See [[w:Hebephilia|Hebephilia]] and [[w:Ephebophilia|ephebophilia]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:58, 12 December 2024 (UTC) 376g2k151zx9vdkitcnctgvexj25p4l 6 317204 2691712 2024-12-12T23:05:10Z Young1lim 21186 {{Information |Description=LIB.2A: Shared Libraries (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691712 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2A: Shared Libraries (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} trab395otko5ohk7kg84tiwop6hdy4n 6 317205 2691715 2024-12-12T23:09:49Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691715 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} mw4aerqfuduh1dbu788f7sohd4cbe77 6 317206 2691717 2024-12-12T23:17:26Z Young1lim 21186 {{Information |Description=ARM.2ASM: Branch and Return Methods (20241213- 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691717 wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Branch and Return Methods (20241213- 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} r1teavqvuct2kx4rxpwl7su1sygi0sg 0 317207 2691745 2024-12-13T07:29:07Z Bert Niehaus 2387134 Bert Niehaus moved page [[Holomorphism/Criteria]] to [[Holomorphic function/Criteria]]: move subpage to appropriate parent page "Holomorphic function" 2691745 wikitext text/x-wiki #REDIRECT [[Holomorphic function/Criteria]] 25ajc8846oinqhxgmz38f8opipgxl1o 6 317209 2691772 2024-12-13T09:25:34Z Young1lim 21186 {{Information |Description=Work2.1A: Libraries (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2691772 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Libraries (20241213 - 20241212) |Source={{own|Young1lim}} |Date=2024-12-13 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} nf04o2bbrhz3p8pxtyqowdcydgeb6np